STOCKHOLM SWEDEN 2020,
Adaptive Trajectory Tracking
Control of a UAV using Gaussian Processes
PHILIPP ROTHENHÄUSLER
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE
Adaptive Trajectory Tracking Control of a UAV using Gaussian
Processes
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Student Philipp Rothenh¨ausler
phirot@kth.se Supervisors Jieqiang Wei, Frank Jiang jieqiang@kth.se, frankji@kth.se
Examiner Karl Henrik Johansson
kallej@kth.se
Monday 27
thJanuary, 2020
Abstract
Unmanned aerial vehicles are a popular choice for various transport and mon- itoring applications. In many outdoor applications frequent variations of the vehicle configuration and time-varying disturbances originating from the en- vironment are expected. In order to counteract resulting deviations from a specified trajectory both model-based and data-driven methods are utilised.
With the performance expectations encoded in a designer chosen trajectory, the implementation of a nonlinear controller based on geometric methods on the nonlinear Lie group SE(3) guarantees stability for the nominal system. In combination with an adaptive compensation term using a radial basis func- tion network (RBFN) and a learning-based term using a Gaussian Process, a learning-based framework is proposed. The approach is evaluated with respect to its real time applicability and numerical results are provided for the adaptive RBFN control approach.
Sammanfattning
Obemannade luftfarkost ¨ar ett popul¨art val f¨or olika typer av transports och
¨overvaknings applikationer. I m˚anga utomhus till¨ampningarna, f¨orekommer det ofta variationer av fordonets konfiguration och tidsvarierande st¨orningar fr˚an omgivning. F¨or att motverka avvikelse fr˚an en ¨onskade k¨orbana, b˚ade mod- ell baserade och data drivna metoder ¨at till¨ampade. Med prestandaf¨orv¨ant- ningarna inb¨addade i en vald bana, stabilitet f¨or det nominella systemet ¨ar garanterad med implementering av en icke-linj¨ara regulator baserad p˚a ge- ometriska metoder p˚a den icke-linj¨ara Lie-gruppen SE(3). I kombination med en adaptiv kompensationsterm som anv¨ander ett radiellt basfunktionsn¨atverk (RBFN) och en maskininl¨arning baserad term med anv¨andning av Gaussian Pro- cess, ett inl¨arningsbaserat ramverk ¨ar f¨oreslaget. Metoden ¨ar utv¨arderade med avseende p˚a dess realtids till¨ampbarhet och numeriska resultat tillhandah˚alls f¨or den adaptiva RBFN styrningsmetod.
Foreword
For the consumption of this thesis it may be helpful to annotate a few things.
Most of the proofs and more extensive derivations are moved to the appendix in order to improve the reading flow.
The mathematical colorboxes are included to provide mathematical intuition by stating assumptions or definitions in green, proved statements in blueish and examples or elaborations in red.
Furthermore the common notations are such that vectors are bold with v and matrices capitalised with M . The only devitation from this notation are applied operators on vectors or matrices such as the hat map ˆv or the vee map M∨.
List of Figures v
List of Tables vii
List of Abbreviations vii
List of Symbols ix
1 Introduction 1
1.1 Motivation . . . 2
1.2 State of the Art . . . 4
2 Preliminaries 10 2.1 Attitude Parametrisation . . . 11
2.2 Lyapunov Theory . . . 20
2.3 Adaptive Control . . . 23
2.4 Gaussian Processes . . . 26
3 Model 29 3.1 Aerial Vehicle . . . 30
3.2 Actuators . . . 30
3.3 Dynamics . . . 34
4 Uncertainty 37 4.1 Endogenous . . . 38
4.2 Exogenous . . . 40
iii
5 Trajectory Tracking 41
5.1 Reference Trajectory . . . 42
5.2 Nominal Geometric Control . . . 44
5.3 Lyapunov Redesign . . . 46
5.4 Adaptive Control using RBFN . . . 56
5.5 Adaptive Control using GP . . . 62
6 Numerical Results 64 6.1 Toolchain . . . 65
6.2 Nominal . . . 65
6.3 Off-Nominal . . . 69
7 Conclusion and Future Work 91 8 Appendix 94 8.1 Algorithms . . . 95
8.2 Annotations . . . 100
8.3 Proofs . . . 102
2.1 Representation of Euler Angle Rotation Sequence . . . 14
2.2 Model reference adaptive control architecture. . . 26
3.1 Aerial vehicle airframe. . . 30
3.2 Illustration of rotor geometry. . . 33
3.3 Illustration of kinematics and kinetics of aerial vehicle. . . 35
5.1 Adaptive RBFN trajectory tracking control framework . . . 56
5.2 Cascaded adaptive RBFN control framework for matched uncer- tainty . . . 56
5.3 Cascaded adaptive RBFN control framework for unmatched un- certainty . . . 60
5.4 Bayesian hyperparameter optimisation for isotropic kernel function. 62 5.5 Adaptive trajectory tracking control framework using GPs . . . . 63
6.1 Nominal system performance for different maneuvers. . . 66
6.2 Nominal system error states for circular trajectory. . . 67
6.3 Nominal system states for circular trajectory. . . 68
6.4 Nominal system control signals. . . 69
6.5 Nominal system error states under endogenous disturbance. . . . 71
6.6 Adaptive RBFN control system error states under endogenous disturbance. . . 72
6.7 Adaptive RBFN control signals for endogenous disturbance. . . . 73
6.8 Adaptive RBFN weight convergence for endogenous uncertainty. 73 6.9 Nominal system error states under exogenous thrust vector un- certainty. . . 75
v
6.10 Nominal geometric trajectory tracking control system 2d and 3d trajectory under globally persistent matched exogenous uncertainty. 76 6.11 Adaptive RBFN control system error states under exogenous
thrust vector uncertainty. . . 78 6.12 Adaptive RBFN control system 3d trajectory under exogenous
thrust vector uncertainty. . . 79 6.13 Adaptive RBFN control system error states under exogenous
thrust vector uncertainty. . . 80 6.14 Nominal system error states under exogenous thrust vector un-
certainty. . . 81 6.15 Nominal system states. . . 82 6.16 Adaptive RBFN matched control system error states under ex-
ogenous thrust vector uncertainty. . . 83 6.17 Adaptive RBFN matched control system 3d trajectory under ex-
ogenous thrust vector uncertainty. . . 84 6.18 Adaptive RBFN matched control system error states under ex-
ogenous thrust vector uncertainty. . . 85 6.19 Adaptive RBFN matched control system error states under ex-
ogenous thrust vector uncertainty. . . 86 6.20 Adaptive RBFN matched control system error states under ex-
ogenous thrust vector uncertainty. . . 87 6.21 Adaptive translational thrust vector RBFN control system weights
for constant unit basis functions under globally persistent exoge- nous unmatched disturbance. . . 89 6.22 Adaptive translational thrust vector RBFN weights for 10 basis
functions for exogenous unmatched uncertainty . . . 90 6.23 Adaptive translational thrust vector RBFN weights for 100 basis
functions for exogenous unmatched uncertainty . . . 90
6.1 Nominal nonlinear geometric control parametrisation . . . 65 6.2 Adaptive Translational RBFN under endogenous uncertainty . . 70 6.3 Global exogenous uncertainty parametrisation . . . 74 6.4 Parametristation of adaptive translational RBFN under globally
persistent exogenous matched uncertainty . . . 77 6.5 Parametristation of adaptive rotational RBFN under globally
persistent exogenous matched uncertainty . . . 77 6.6 Adaptive translational thrust vector RBFN with 10 basis func-
tions under exogenous unmatched uncertainty . . . 88 6.7 Adaptive translational thrust vector RBFN with 100 basis func-
tions under exogenous unmatched uncertainty . . . 89
vii
v Lower case, bold notation for some vector v.
M Uppercase for some matrix M . Rn n-dimensional real space.
SO(n) Special orthogonal group with SO(n)⊂ Rn×n. Sn n-sphere with Sn⊂ Rn+1.
so(n) Group of skew symmetric matrices ˆω∈ so(n) ⊂ Rn×n. ˆ
v Skew symmetric matrix of a vector v.
M∨ Vector from skew symmetric matrix M .
vB Vector with coordinates relative to some frameB.
RAB Rotation matrix representing frame transformation from some frameB to some frame A.
cη Cosine function with η in radians.
sη Sine function with η in radians.
exp Exponential map exp transforming some skew symmetric matrix ˆ
ω∈ so(3) into a rotation matrix R ∈ SO(3).
kvk Euclidean vector norm withkvk ≡ kvk2. (2-norm) vu Some unit vector normalised with vu= vkvk−1.
tr(M ) Trace operator summing diagonal elements of some matrix M . p Translational position state in inertial frameI coordinates.
viii
v Translational velocity state in inertial frameI coordinates.
R Attitude of aerial vehicle’s body frameB relative towards inertial frameI.
ω Angular velocity state in inertial frameI coordinates.
ξ Partitioned state vector with ξ = (p, v, R, ω)T.
λη Maximum eigenvalue of some matrix η according to max
λ λ(η).
λη Minimum eigenvalue of some matrix η according to min
λ λ(η).
Introduction
In this chapter, the target platform and the motivation for this work is in- troduced. First, the motivation is detailed with a focus on model-based and data-driven control approaches. Subsequently, an outline of the structure for this work is provided.
Following, the state-of-the-art for possible aerial vehicle applications, modeling approaches, trajectory generation methods and control frameworks are stated.
Here, the latter is treated distinctively for both conventional and learning-based methods.
1
1.1 Motivation
Introduction
In this work, an adaptive trajectory tracking control approach for an aerial vehicle is developed. The control framework compensates for endogenous dis- turbances originating from model uncertainty as well as matched exogenous disturbances. The target platform is a quadrotor with a racing drone airframe, which is introduced in more detail in Section 3.1.
The aerial vehicle dynamics are modelled using the Euler-Newton approach.
The attitude is parametrised with a coordinate-free attitude representation us- ing rotation matrices. Together with a geometric trajectory tracking control that is almost globally exponentially attractive, the nominal nonlinear trajec- tory tracking control is designed for analytic trajectories that are at least three times differentiable. With this nominal trajectory tracking control, possible model uncertainties (endogeneous) and external disturbances (exogenous) are assumed and the resulting deviations evaluated in simulation. In order to coun- teract the resulting deviation from a designer specified trajectory, the bounded- ness of the uncertainty is exploited with a robust control approach using Lya- punov redesign. Subsequently, an adaptive control approach using a radial basis function network (RBFN) with preallocated centers is used. Following from the disadvantage in form of domain restrictions, the adaptive control approach is limited on previously encountered environments and configurations.
In order to extend the locally successful disturbance rejection using adaptive control, a possible learning based framework using Gaussian processes is pro- posed that is aimed to cover a global domain.
The adaptive control algorithm using a RBFN is implemented in a Python- based simulation. The development and evaluation is conducted with focus on their real time applicability.
Motivation
Aerial vehicles such as quadrotors and hexrotors have spread across various disci- plines in industry, research and hobbyist fields. Several use-cases are illustrated in the subsequent Section 1.2.
One of the main motivations for this thesis is the use-case of naval monitoring of algae coverage in the pacific[1]. In maritime environments, time-varying wind disturbances are expected. In addition, it is expected to experience uncertainties originating from different vehicle configurations such as varying sensor equip- ment. Hence, a trajectory tracking control insensitive to model uncertainties is desired.
The basis for the development of a tracking controller is the decision for either a model-based control (MBC) or a data-driven control (DDC) approach. Choos- ing the former approach requires system identification based on data or using first principles to achieve a model of the plant. This implies the “certainty equivalence principle”, namely that the model accurately represents the true system [2]. Since model mismatch is present in the identification process at all times, there is a necessity in the MBC approach for subsequent robust control theory or adaptive control. Therewith, the basis of MBC theory is the “cer- tainty equivalence principle” and, hence, a successful practical implementation is strongly tied to a well designed model. Consequentially, the design process has to balance accuracy and adequate computational complexity for a successful implementation on the target platform. In contrary, the data-driven control ap- proach is purely based on measured input/output data of the plant and does not include equally strong assumption about the plant’s dynamics. This seems to be especially applicable to more complex plants that are uncommon or difficult to model.
Since, however, the quadrotor has many well established modelling techniques [3]
and, consequently, well-established system identification procedures, the choice of a MBC approach was made. The expected model mismatch is shown to satisfy the matching condition and subsequently counteracted using Lyapunov redesign.
Following, an adaptive control approach is implemented and numerical results provided. Based on the results a learning-based framework using a Gaussian process (GP) is proposed.
Outline
In the first chapter, the motivation for this work is given and current state- of-the-art for drone applications, modelling, trajectory generation methods are described. Subsequently, conventional control frameworks and possible machine learning algorithms are illustrated in more detail.
In the second chapter, the preliminaries for this work are introduced. Initially, a careful outline for various attitude parametrisation methods are stated. Fol- lowing, the essential Lyapunov theory concepts are provided. Finally, both elementary concepts for adaptive control and Gaussian processes are briefly in- troduced.
In the third chapter, the modelling process of the aerial vehicle is motivated and the nonlinear dynamics are introduced.
Following, in the fourth chapter, the considered uncertainty cases are stated.
From an initial derivation based on the modeled dynamics parametric uncer- tainty in form of a parameter space is given. As a consequence, it is proven that the parameter space can be represented as a matched uncertainty, which subsequently is further exploited. In addition, an analytically defined exogenous force is introduced for further numerical analysis.
In chapter 5, the nominal tracking controller is stated and a corresponding analytic trajectory tracking reference generation algorithm provided. Following, the derived matched unstructured uncertainty for model mismatch is exploited and the stability conditions for the robustness based on the Lyapunov redesign control approach are derived. Subsequently, the stability conditions for the parametric adaptive control approach using a radial basis function network are derived. Finally, a data-driven control framework is proposed to compensate previously encountered disadvantages of conventional control frameworks.
In chapter 6, a Python based simulation framework is employed to demonstrated the conceptual performance of the nominal control. Furthermore, numerical results are provided for the uncertainty cases for both endogenous and exogenous disturbances.
In chapter 7, a conclusion and outline of future work is proposed.
1.2 State of the Art
1.2.1 Applications
The target platform for this work is an aerial vehicle which constitutes the min- imal configuration of rotors, namely a quadrotor. The platform is commonly chosen for various control implementations. Some of the conventional meth- ods include sliding mode control, model predictive control, geometric control and extensions based on robust control theory. Adaptive control methods in- clude adaptive pole placement methods and machine learning methods using for example neural networks to compensate for model uncertainty.
Subsequently, both industrial, research and hobbyists use-cases are introduced.
In the industrial context, two of the leading manufacturers Ascending Tech- nologies and DJI provide various use-cases, including inspection and monitor- ing tasks, surveying and mapping and aerial imaging [4] or other industrial services such as offered by DJM-aerial [5]. Industrial use-cases also include the exploration of mining facilities with mapping algorithms, such as Kespry and ABB [6]. In addition, the transport of parcels and emergency medicine between hospitals is becoming an emerging application of quadrotors [7].
Further use-cases are delivery of parcels, real time inspection of industrial as- sets or large scale manufacturing facilities. Similarly, the demand of scanning agricultural production sites initiated also the use of drones for items in ware- houses. All of these applications allow to conduct inspections cheaper and more frequent with less safety hazard for the inspectors [8].
For hobbyists, the most prominent use-case is recording video footage and ex- ploring environments using telemetry. Furthermore, the new discipline of drone racing has an increasing audience. Examples for these competitions can be
found in [9] and in [10], where the first is a manually controlled competition and the latter a competition based on autonomous algorithms.
1.2.2 Modeling
A common modeling approach is applying the Euler-Newton method as de- scribed in [3]. In addition, in [11] several aerodynamic effects are considered such as drag forces and gyroscopic effects. With these, the model based con- trol approach becomes more thorough and allows for high performance tracking control operation without the need of robustification or iterative learning con- trol. Furthermore, several commonly made assumptions include the rigid and symmetric body assumptions and considering the cog in the origin of the body frame. This work also offers intuitive parameter identification methods such as the static thrust test. In addition, the performance of the nonlinear model is demonstrated by employing a nonlinear model predictive controller in sim- ulation. In [12], the assumption of the motor model as a first order system is made and the saturation limits are approximated via experimentation. In addi- tion, the inertia matrix J is found in [12] by constructing a physically accurate Solidworks model.
Another common challenge in the control of aerial vehicle’s is the consideration of the ground effect. In the vicinity of 2 propeller diameters strong aerody- namic disturbance is observed. In [13], the dynamics for a learning based model predictive control is used to adapt to these additional dynamics within seconds.
In [14], a quaternion-based model is assumed for the attitude control of a quadro- tor. Additionally, the motor dynamics are modelled as a first order dynamical system following [15]. The resulting complete model admits also aggressive ma- neuvers. This shows that for highly dynamic maneuvers the consideration of the rotor dynamics are essential. In [15] the same model is applied and linerisation about the hover point of the vehicle is used for the control design.
After evaluation of the corresponding system identification tasks required for more thorough nonlinear models of quadrotors, the classical Euler-Newton model was chosen for this work. This choice is especially motivated with hindsight towards the desired learning based approach in addition to the initially imple- mented nominal control. Recent work that uses this model can be found in [16, 17,18].
1.2.3 Trajectory Generation
Imperative for controlling an aerial vehicle is the generation of a feasible trajec- tory. An intuitive approach was initially proposed in 1996 by M.v.Nieuwstadt and R.M.Murray in [19] by proposing the property of differential flatness for some dynamical systems. In this work the trajectory and input generation for
following a spatial path is shown based on geometric principles. Hereby, the generation is purely based on a subset of initial states that can derive so called flat outputs. This method has been adopted in various implementations for the trajectory and feedforward control input generation, such as in [17] and [18]. The former proposes a keyframe based trajectory generation algorithm using n-th order polynomial interpolation. Based on minimising a functional in terms of a quadratic program, the corresponding quadruplet for each keyframe is found. Following, the resulting trajectory is tracked using the nonlinear con- troller in [16]. In the implementation, the feedback terms originating from the Euler-Newton model and feedforward terms for angular velocities are neglected.
For feasibility considerations the relation of the derivatives to the control in- puts is shown. Similarly, in [18], the trajectory generation is derived based on differential flatness and corresponding feedforward terms are deduced from its principles. The necessary flat outputs are achieved by partial differentiation of a logarithmic spiral expressed in spatial coordinates. For the tracking task the nonlinear controller in [16] is used. However, no feasibility guarantees are provided in this implementation. But in addition to the conventional trajectory generation, an obstacle avoidance algorithm is presented based on a potential field alike approach that adjusts the previously generated vector field to repel the aerial vehicle from the obstacle’s position. In both cases experiments of the implementation are provided.
In [20], strong focus has been put on real-time trajectory generation, feasibil- ity, and optimality. The limitations regarding thrust and angular body rates are considered and the derived optimal trajectory generation algorithm is also suggested as a feedback law.
While investigating the trajectory generation approach also the concept of model reference adaptive control (MRAC) appears frequently, given that in the nature of this control concept the reference generation is meant to define a certain plant behaviour as a trajectory tracking reference. However, the question of feasibility and optimality for trajectories often remains unanswered in the literature. For example in [21], a piece-wise constant command signal with a subsequent first order low-pass generates a continuous reference signal. However, no considera- tions of limited control authority for the rotors is made. In [17], however, the safety consideration is included by scaling the interpolation solutions by some factor that reduces the dynamic requirements.
1.2.4 Control
To counteract possible disturbances the approach in [16] is extended in [22]
using PID control, in [23, 24] using a robust adaptive control with σ- and - modification, providing bounded solution trajectories. By defining a first body axis that converges only to the projection to the plane perpendicular to the body frame z-axis, [25] extends the method of the differential flatness based trajectory generation approach and therewith avoids some otherwise arising singularity.
In [14] a disturbance observer-based (DOB) control approach is employed for aggressive maneuvers such as double loops. The performance is demonstrated in outdoor experiments as well as indoor experiments. In [15], a linearised model about the hover point is basis for a backstepping alike control approach. The implementation of the position control is separated into an onboard and offboard control loop, where the onboard attitude control loop iterates at 1kHz. External commands are transformed to the vehicle with a loop frequency of 100Hz.
1.2.5 Learning-based Control
A common approach to extend the previously mentioned classical control ap- proaches, is to embed learning-based methods to compensate for the shortcom- ings in form of domain restrictions or disturbance sensitivity.
The choice of available learning-based methods are amble and in many ways closely related with only subtle nuances to differ from each other. With this ambiguity and the fast paced evolution of methods, a lacking methodology leads often to a more heuristic approach by selecting approaches that are commonly used in the desired field of application.
For quadrotors, common choices of learning-based methods range from neural networks, over reinforcement learning to probabilistic approaches.
In most cases the setting for the approach is to estimate some model uncer- tainties. In [26], a reinforcement approach is applied for a end-to-end control approach. With commonly used approaches such as neural networks or rein- forcement learning, either labeled data for learning or a multitude of iterations is required. The learning process is driven by data and hence the approaches are labeled data-driven. In the learning process usually gradient-based methods are applied and the employed optimisation algorithms range from stochastic gra- dient descent (SGD) descent1to adaptive moment estimation (Adam). While these methods can show fast convergence, implicit problems include local min- ima and the necessitiy to apply methods for the avoidance of overfitting such as for example cross-validation. In addition, the conventionally applied frame- works don’t provide probabilities about the uncertainty estimate and therewith making it difficult to evaluate the performance of the approach.
In [25, 27, 28], a neural network is applied to compensate uncertainties in a model-based approach using the controller based on geometric methods in [16].
In this approach Lyapunov theory based stability arguments are derived consid- ering the neural network with a sigmoid activation function.
In [29], a hybrid model reference adaptive control approach is proposed with learning approach applied. First a concurrent learning algorithm identifies un- matched uncertainties, then after its convergence a adaptive controller updated with an estimate of the unmatched uncertainty guarantees convergence to the
1Also known as backpropagation.
reference trajectory under additional matched uncertainty. The reference is linear and the convergence of the unmatched uncertainty is guaranteed even without persistency of excitation (PE) given some assumptions on a history of saved state measurements.
With the advancing computational power of embedded devices and improved GPU facilitation in available probabilistic libraries, Gaussian processes and probabilistic approaches in general gain more attention in the machine learning community. In contrary to conventional methods, Gaussian processes are at- tractive due to their capability to act as generative models and the implicitly available probabilities for generated estimates. In recent work Gaussian pro- cesses are applied to various platforms, some of which are miniature race cars with the cautious nonlinear model predictive control (NMPC) [30, 31] or in [32] to the problem of robotic manipulation by combining backstepping with a Gaussian process.
The modern control theory approaches introduced in the Section 1.2.4 are based on a rich history and provide strong stability arguments. With their nature lay- ing in model-based methods, a usual requirement is the certainty-equivalence principle and hence the necessity for an accurate physical modeling process. In contrary the machine learning approaches are dominated by heuristics and de- rive motivation for method selection from experience as an established method- ology. By utilising the flexibility of heuristics driven approaches from the ma- chine learning community and well established and more principled approaches from the modern control theory community, hybrid controllers follow.
With the combination of both modern control theory approaches such as adap- tive or robust controllers and learning-based methods such as Gaussian pro- cesses, the resulting hybrid controllers yield a principled approach to machine learning challenges. In [33], an adaptive control approach is combined with a Gaussian process. The adaptive controller acts as a generative model providing dynamic estimates for the Gaussian process. Thereby utilising the error-driven nature and fast convergence of adaptive control and the noise compensation resulting from the probabilistic approach of the Gaussian process. Noteworthy in this approach is also the strong emphasis on the challenges to implement Gaussian process based controllers due to the curse of dimensionality. In or- der to counteract this disadvantage, the proposed solution is a budgeted kernel approach referencing previous work in [34].
In [35,36], a model predictive controller is combined with a Gaussian process.
By doing so the uncertainties estimated by the Gaussian process are considered in a directy way and reduces computational burden in comparison to existing GP based stochastic MPC approaches.
In [37], an adaptive control framework using Gaussian processes with online hyperparameter estimation is proposed. In the framework the emphasis is put on the importance of hyperparameter optimisation, which is done using a modified likelihood function.
Since in the Gaussian process based approach, the kernel function encodes the nature of the underlying uncertainty, in [38] an extensive and detailed depic- tion of kernels and their superposition for deriving suitable kernel functions is provided. However in the majority of currently employed approaches the conventional isotropic radial basis function kernel is selected. In some more targeted approaches the automatic relevance determination (ARD)2 kernel is chosen and therewith using distinct lengthscales for each input dimension, re- spectively feature.
In the tutorial [39], a detailed background and derivation of GP is provided.
Advantages of GPs and possible disadvantages are stated. For the challenge of real time applicability due to the curse of dimensionality, a budgeted kernel approach is referenced. Another approach labeled sparse online GP algorithm efficiently approximates the Kullback-Leibler (KL) divergence, where the dat- apoint with the largest KL divergence is deleted. and currently available im- plementation libraries are stated. For the latter the Python library GPy [40] is both efficient and provides advanced algorithms such as multiple outputs. Other common libraries include PyMC3 [41] for Python and Stan [42] for c++. Where the latter two libraries both provide the efficient NUTS sampling algorithm, based on Hamilton Monte Carlo. Originally multiple outputs in the standard GP framework required multiple outputs to be independent, which can lead to greatly deteriorated estimates given sparse data, see [39, p. 81]. With [43], a method for constructing non-independent coupled multidimensional output Gaussian processes is provided. Related an interesting summary of multidi- mensional Gaussian processes can be found in [44].
2Also known as sparse bayesian learning.
Preliminaries
In the following chapter the common attitude parametrisation methods such as Euler angles, rotation vectors, quaternions and rotation matrices are introduced.
Subsequently, elementary definitions for the application of Lyapunov theory are provided. In similar fashion useful notions for adaptive control and Gaussian processes are briefly detailed.
10
2.1 Attitude Parametrisation
The initial assumption with the aerial vehicle constituting a rigid body, leads to the description of the vehicle state in the nonlinear configuration space using the special Euclidean Lie group SE(3)[45]. The representation then follows as a tuple of (p, R)∈ R3× SO(3). Here p describes the position and R the attitude of the rigid body. The latter is defined using a rotation matrix and has some unique properties which will be more thoroughly introduced in the subsequent sections.
The nonlinear nature of the configuration space of SE(3) is attributed to the nonlinear special orthogonal Lie group SO(3) describing the attitude in R3 on the manifold of the unit 2-sphere S2. Given the attitude evolving in nonlinear space, the parametrisation representing this attitude becomes a crucial design choice in the development of control systems.
Commonly, a local parametrisation with a triplet (ϕ, ϑ, ψ)∈ R3 of Euler angles is used. Each element represents a rotation around one coordinate axis. Hence, this parametrisation is mostly chosen due to its intuitive nature in understanding the body’s attitude. Since however the Euler angles cannot represent the whole 2-sphere and have singularities (gimbal lock ), it is necessary to apply domain restrictions for the control system. Alternatively, the representation deficiencies can be counteracted by re-parametrisation methods such as e.g. the Intra-Euler- Angle Conversion[46].
In order to avoid these disadvantages, the global parametrisation using a quadru- ple (q0, qx, qy, qz)∈ R4 can be chosen. Under some special notation and addi- tional assumptions these are called unit Quaternions (versors) with q ∈ H.
Here, the attitude is specified by a single rotation axis vector and a scalar defining the rotation angle around this axis. The geometrically very intuitive interpretation of the attitude motivates to choose this parametrisation. Further- more, the representation does not contain any singularities, which is beneficial for the control system design. While this clearly motivates attitude parametri- sation using quaternions, the additional dimension brings the disadvantage that unit quaternions represent the unit 3-sphere S3, which double covers the unit 2-sphere. Hence, each attitude is represented by two antipodal quaternions. In order to guarantee performance of the attitude control system, it is required to choose the correct quaternion of the antipodal tuple and then generate the trajectory on the manifold such that it coincides with the shortest rotation.
With the coordinate-free attitude representation using a 9-tuple (r1, r2. . . , r9)∈ R9 in form of a rotation matrix R∈ SO(3) ⊂ R3x3, global singularity free and unique attitude parametrisation on the 2-sphere is achieved. While this is clearly desirable, the additional dimensions constitute a major drawback for real time implementations using model based control techniques. Furthermore, the lack of intuitive geometrical interpretability and its sensitivity to rounding errors in numerical applications adds additional disadvantages.
In the subsequent sections the various attitude parametrisation methods are de- tailed and the generation of a rotation matrix from their corresponding compo- nents is described. For the quaternion and the coordinate-free parametrisation, also its differential equations are shown. Therewith providing a reference for future implementations in dynamical systems.
2.1.1 Coordinate Frames
Preliminary for the representation of a rigid body’s attitude is the definition of a reference frame. Usually this reference frame is considered as the inertial frame I.
In aviation a common convention is the so called north-east-down (NED) refer- ence frame, which is motivated by the fact that the point of interest usually lies below the aerial vehicle. This frame orients the x axis tangent to the latitude towards east, the y axis tangent to the longitude towards north and the z-axis geocentrically towards the earth and hence orthogonal to the xy-plane.
In robotics and terrestrial vehicles the common convention is the east-north-up (ENU) reference frame. Which corresponding to its name, orients the x-axis along the east direction and the y-axis along the north direction of the local xy- plane. Similarly, the z-axis is orthogonal to that plane and pointing in opposite direction of the vector from the aerial vehicle geocentric towards the earth.
With the target of external control utilising a robotics middleware in this work the ENU frame is considered.
Coordinate Frame Transformations
Ultimately, the goal from each attitude parametrisation is the capability to transform a vector ~a ∈ R3 represented in coordinates with respect to some frameB into the same vector aI ∈ R3 represented in coordinates with respect to some frameI.
Hence, the definition of a rotation matrix is natural. Subsequently, the prop- erties of this global singularity-free and unique parametrisation for elements on the n-1-sphere Sn−1and in specific for attitudes in R3the 2-sphere S2is detailed.
The group of all rotation matrices Rn×ndefined by
SO(n) ={R ∈ Rn×n| RRT = I, det R = 1},
is labeled the special orthogonal group and for the case n = 3 with SO(3) the rotation group of R3.
The rotation group SO(3) defines the configuration space of a freely rotating rigid body relative to a fixed frame. Furthermore a rotation matrix RAB ∈ SO(3)
represents a transformation of a vector vB with coordinates relative to frameB into a vector vAwith coordinates relative to frameA.
Corresponding to Equation (2.1.1) the transformation from coordinates of frame A, to coordinates of frame B can be achieved using the inverse rotation matrix RBA= R−1AB= RTAB.
Let the space of all skew symmetric matrices Rn×nbe defined as so(n) ={S ∈ Rn×n| ST =−S}.
Proposition 2.1: Hat Map
Let the hat map with wedge operator ∧ and its corresponding element be denoted with ˆ· and
(·)∧: R3→ so(3), so that
(a)∧= ˆa
=
0 −az ay
az 0 −ax
−ay ax 0
.
With a∈ R3 and ˆa∈ so(3) the following relation holds a× b = ˆab = −ˆba.
Lemma 2.1: [45] Hat Map Properties
Given R∈ SO(3) and v, ∈ R3, the following properties hold:
R(v× w) = (Rv) × (Rw), R(w)∧RT = (Rw)∧.
2.1.2 Local Coordinates
According to Euler’s rotation theorem each rotation can be represented by three angles (ϕ, ϑ, ψ). Here each angle corresponds to a rotation about one of the principal axes of the frame with ϕ about the x-axis, ϑ about the y-axis and ψ about the z-axis. The combination of these principal rotations provides various parametrisation. An intuitive and detailed introduction to Euler angles and its relation to rotation vectors1can be found in [47,48]. Note that in the definition
1Also denoted as Euler parameters and under given corresponding notation Quaternions.
Figure 2.1: Representation of Euler angles rotation sequence in a North-East-Down (NED) coordinate frame[48, Fig. 4.8].
of Euler Angles it is also often distinguished between proper Euler Angles and Tait-Bryan Angles. Here the latter are about three distinct axes (xyz) and the former have the same axis for the first and last rotation (xyx). Essentially the attitude representation using Euler angles is achieved by a sequence of three rotations about the principal (coordinate) axes.
As illustrated in Figure 2.1, the exemplary rotation sequence is a Tait-Bryan angle zyx-sequence. This sequence is described by a initial rotation around the z-axis with angle ψ, a subsequent rotation around the new body frame y- axis with angle ϑ and a final rotation about the newly generated x-axis with angle ϕ. The rotation about the new coordinates frames after each rotation is called intrinsic rotation sequence, where the rotation about a inertial coordinate system is called extrinsic.
Definition 2.1: Principal Rotations
The principal rotations for a Euler Angle sequence are defined by
Rϕ=
1 0 0
0 cϕ sϕ
0 −sϕ cϕ
,
Rϑ=
cϑ 0 −sϑ
0 1 0
sϑ 0 cϑ
,
Rψ=
cψ sψ 0
−sψ cψ 0
0 0 1
where cη = cos(η) and sη= sin(η). This yields the definition of a rotation matrix Rϕϑψ from the Euler angles (ϕ, ϑ, ψ) as follows.
The rotation matrix for the Euler angle sequence zyx is composed of the prin- cipal rotations
Rϕ,ϑ,ψ= RϕRϑRψ
and yields the final rotation matrix R∈ SO(3) with
Rϕ,ϑ,ψ=
cψcϑ sψcϑ −sϑ
(cψsϑsϕ)− (sψcϕ) (sψsϑsϕ) + (cψcϕ) cϑsϕ
(cψsϑcϕ) + (sψsϕ) (sψsϑcϕ)− (cψsϕ) cϑcϕ
.
The Euler angles for the zyx-sequence can be retrieved from the rotation matrix R with
ϕ = tan−1(r23, r33), ϑ =− sin−1(r13), ψ = tan−1(r12, r11),
(2.1)
where ri,j is the element of the i-th row and j-th column of R.
Clearly the generation of the rotation matrix requires the use of numerous trigonometric functions. In addition this parametrisation has the deficiency of the so called gimbal lock. This singularity in the representation happens when the rotation axes of rotation sequences coincide and the representation losses a degree of freedom. In the zyx-sequence this corresponds to the x axis pointing upright, hence (ϕ, ϑ, ψ) = (η1, π/2, η2) for any η1, η2∈ [0, 2π). Since Rψ around some inertial z-axis coincides with the rotation around the body frame x-axis.
This is due to Rϑaligning the body frame x-axis in exactly the same orientation to the inertial axis corresponding to the initial configuration described by body frame’s z-axis. Hence, the rotation using Rϕ covers the same segment of the
2-sphere S2 as the rotation Rψ. Analytically, this can be seen from Equation (2.1), since under the zyx-sequence all angles ψ = η1and ϕ = η2with ϑ = 2/π generate a rotation matrix that yields (ϕ, ψ) = (0, 0) in its inverse mapping from Equation (2.1). Hence, the mapping is many angles to one rotation matrix and not invertible [47].
Adjusting the rotation sequence between the 12 possible ways shifts this sin- gularity to different points in the triplet (ϕ, ϑ, ψ), but never removes it.2 This deficiency often motivates the choice for global coordinates instead of the local Euler angle coordinates on the manifold S2.
2.1.3 Rotation Vectors
With the initially mentioned disadvantages of the Euler angles, alternative rep- resentations are desired. In fact the sequence of rotations about each coordinate axis can be summarised as a rotation about a fixed axis as stated by Euler in Theorem 2.1.
Theorem 2.1: [45] (Euler) Angle Axis Theorem
Any orientation R∈ SO(3) is equivalent to a rotation about a fixed axis ω∈ R3 through an angle θ∈ [0, 2π).
This yields the so called equivalent axis representation3. In order to generate a rotation matrix from the tuple (ω, θ), the exponential map can be used.
The tuple can be easily found for a desired rotation from two vectors with a∈ R3 to ~b∈ R3by using
ω =ka × bk−1a× b, θ = aTubu.
Proposition 2.2: Exponential Map
Let the exponential map with ˆω andkωk = 1 with θ ∈ R be defined by exp : so(3)→ SO(3),
so that
eωθˆ ∈ SO(3).
2For a more detailed overview of various Euler angle parametrisations sequences and their conversions between alternative representations see [46].
3Also called angle-axis representation.
Hence R∈ SO(3) can be efficiently computed with the Rodrigues’ for- mula
R = eωθˆ = I + ωˆ
kωksin(kωkθ) + ωˆ2
kωk2(1− cos(kωkθ)).
Following from the Rodrigues’ formula the rotation vector and its angle can be retrieved as follows
θ = cos−1 tr(R)− 1 2
,
ω = 1 2sθ
r32− r23 r13− r31 r21− r12
.
Given that both (−ω, θ) and (ω, −θ) yield the same R ∈ SO(3), it becomes evident that the mapping exp is many-to-one. Hence, it is natural to choose a parametrisation not only depending on the three components defining ω but also its angle θ. Which is a parametrisation also known as Euler parameters[47]
or under given notation quaternions.[48] Subsequently, we will introduce the complex notation for this 4-dimensional parametrisation and label them quater- nions.
2.1.4 Quaternions
For the application of quaternions as attitude representation the quadruple (q0, qx, qy, qz)∈ R4is introduced. Then following the notation of William Rowan Hamilton from 1843, a quaternion is defined by introducing three additional complex basis vectors (i, j, k) such that
i2= j2= k2= ijk =−1 holds. The resulting quaternion q∈ H then is
q = q0+ qxi + qyj + qzk,
where q0∈ R represents the scalar part and the components (qx, qy, qz) =: ¯q∈ R3 the vector part. This invites the compact representation in form of the tuple (q0, ¯q). The notation difference between the basis vectors i, j, k and the subscripts of the components of ¯q is chosen due to the nature of the vector part representing the rotation axis for some attitude in R3 with axes x, y and z.
Correspondingly, a quaternion representing a rotation about the unit vector urot= (0, 0, 1)T with the rotation angle ψ = π/2 is then defined by
q = (cos(ψ/2), sin(ψ/2)urot).
It can be seen that the rotation axis vector takes the place of the vector ¯q in the quaternion.
The fact that the quaternion is defined using ψ/2 as an argument for the trigono- metric functions, is a result of transforming a 3-dimensional vector of R3 using a 4-dimensional representation of the rotation, which requires repeated appli- cation of the same quaternion in its complex conjugated form to cancels some unwanted rotation.
For the rotation of a vector of R3 by some quaternion, additional concepts are necessary.
The addition of a quaternion q with a quaternion p is adding each component such that
q + p = q0+ p0+ ¯q + ¯p.
The complex conjugate of a quaternion q is q∗= q0− ¯q.
The norm of a quaternion q is
kqk2= qq∗= q02+ qx2+ q2y+ qz2. The inverse of a quaternion q is
q−1= q∗ kqk.
Definition 2.2: Hamilton Product
The quaternion multiplication is denoted by pq = q· v = (q0+ ¯q)· (p0+ ¯p)
= q0p0− ¯q· ¯p + q0p + p¯ 0q + ¯¯ q× ¯p.
=
q0 −qx −qy −qz qx q0 −qz qy
qy qz q0 −qx qz −qy qx q0
p0
px
py
pz
.
In order to multiply a vector from R3 with a quaternion in H to apply a ro- tation, so called pure quaternions are generated. For this a zero scalar part is added to the vector in R3 and therewith lifting it into the ring of quaternions vq ∈ H0⊂ H with vq = (0, v) in the following definition.
Definition 2.3: Quaternion Rotation
The quaternion rotation is defined by a triple quaternion product with q∈ H, vq ∈ H0and q∗∈ H such that the rotated vector v0∈ R3 is
v0= qvqq∗= (q0+ ¯q)(0 + v)(q0− ¯q)
= (2q02− 1)v + 2(vTq)¯¯q + 2q0(v× ¯q).
The corresponding rotation matrix Rq ∈ SO(3) follows as
Rq =
2q20− 1 + 2q2x 2qxqy− 2q0qz 2qxqz+ 2q0qy
2qxqy+ 2q0qz 2q20− 1 + 2qy2 2qyqz− 2q0qx
2qxqz− 2q0qy 2qyqz+ 2q0qx 2q20− 1 + 2qz2
.
Note that the order of these triple quaternion rotation sequence qvqq∗ and q∗vqq can be interpreted as a vector rotation and frame rotation respectively[48].
For dynamical systems the dynamical evolution of attitudes under angular veloc- ities are of interest, hence the derivative of a quaternion is defined by definition 2.4.
Definition 2.4: Quaternion Dynamics
The quaternion derivative of a quaternion q is given by
˙ q = q· ω
=
0 −ωx −ωy −ωz ωx 0 −ωz ωy
ωy ωz 0 −ωx ωz −ωy ωx 0
q0
qx
qy
qz
,
where the quaternion q∈ H and the angular rate of the rotation axis is ω∈ R3.
2.1.5 Coordinate-Free
Given the disadvantage of quaternions resulting from the ambiquity of two an- tipodal quaternions for each attitude in R3 a globally unique and singularity free parametrisation is desired.
This can be achieved by introducing a 9-tuple (r1, r2, . . . , r9) ∈ R9 that rep- resents each element in the rotation matrix. This yields the coordinate-free parametrisation as follows.
With the 9-tuple the rotation matrix R∈ SO(3) results intuitively from
R =
r1 r4 r7
r2 r5 r8
r3 r6 r9
.
Its derivative follows correspondingly without re-parametrisation or introducing local coordinates with Definition 2.5.
Definition 2.5: Rotation Matrix Dynamics
Under the angular velocity ω ∈ R3, the attitude representation R ∈ SO(3) evolves according to
R = Rˆ˙ ω.
While the unique representation of attitudes is very beneficial for control appli- cations, the elements of the rotation matrix are highly susceptible to numerical rounding errors. Furthemore, the dimensionality of this parametrisation poses major challenges for real time implementations in model based techniques such as model predictive control applications.
2.2 Lyapunov Theory
In the feedback design of control systems for nonlinear plants it is desired to show guaranteed stable operation. With the application of Lyapunov stability theory insight into the stability properties of the plant can be derived in a principled way. Clearly, this requires the modeled dynamics to accurately enough represent the true plant’s dynamics.
In Lyapunov theory an energy-alike function is used to show the guaranteed de- cay of the system’s energy content in some domain of operation (local )D ⊂ Rn or in the whole state-space (global ) Rn. Under the assumption of the certainty- equivalenceprinciple and usually additional assumptions on signals and param- eters the concepts of stability, asymptotic stability, exponential attractiveness and exponential stability are shown. Furthermore, the systems are usually sep- arated into two categories autonomous and non-autonomous systems. The first category does not have direct dependence on the independent time variable, while the latter has direct time dependence. Hence non-autonomous systems have varying stability properties depending on the initial configuration along the time axis, which introduces the concept of uniform stability.
In addition, some control systems lack the desirable properties of convergence to the origin as time progresses towards infinity. For these systems it often can be shown that the solution remains bounded around the zero-equilibrium.4.
4This generalisation can be done w.l.o.g. by a simple change of variables, see [49, p.112]
The results in this chapter can be found in more detail in Hassan K. Khalil’s Nonlinear Systems [49] for nonlinear control systems in general and similarly in Alberto Isidori’s Nonlinear Control Systems [50]. For an adaptive control oriented perspective with focus on non-autonomous systems Nhan. T. Nguyen’s Model-Reference Adaptive Control[51] provides ample theoretic and application examples.
2.2.1 Autonomous Systems
Consider the state-space x∈ D ⊂ Rn with the dynamics
˙
x = f (x), (2.2)
where f :D → Rn is locally Lipschitz. We then show stability for the origin x = 0 such that f (x) = f (0) = 0.
Definition 2.6: /δ- Stability The equilibrium x = 0 of (2.2) is
• stable if, for each ε > 0, there is a δ = δ(ε) > 0 such that kx(0)k < δ ⇒ kx(t)k < ε, ∀ t > 0
• unstable if it is not stable
• asymptotically stable if it is stable and δ can be chosen such that
kx(0)k < δ ⇒ limt→∞x(t) = 0
With the definition of basic stability concepts we can now determine Lyapunov functions V (x) according to the following theorem to guarantee stability and asymptotic stability.
Theorem 2.2: [49] Asymptotic Stability
Let x = 0 be an equilibrium point for the system in Equation (2.2) and D ⊂ Rnbe a domain containing x = 0. Let V :D → R be a continuously differentiable function such that
V (0) = 0 and V (x) > 0 inD/{0}
V (x) 6 0 in˙ D Then, x = 0 is stable. Moreover, if
V (x) < 0 in˙ D/{0}, then x = 0 is asymptotically stable.
Definition 2.7: [52] Exponential Attractiveness
An equilibrium point x = 0 of a dynamic system is exponentially attrac- tive if, for some δ > 0, there exists a constant α(δ) > 0 and β > 0 such that
kx(0)k < δ ⇒ kx(t)k 6 α(δ)e−βt ∀t > 0.
Note, that the notion of exponential attractiveness is weaker than exponential stability, in which the rate of convergence depends on the initial configuration withkx(t)k 6 αkx(t0)k.[16]
2.2.2 Nonautonomous Systems
In case of a nonautonomous system with the state-space x∈ D ⊂ Rn and the dynamics
˙
x = f (t, x), (2.3)
the initial conditions affect the stability properties of the plant. This introduces the concepts of uniform stability. For a detailed treatment references can be found in [49,53,51].
In the context of robust control, the notion of boundedness arises. Similarly, adaptive control systems with the persistency of excitation requirement result in bounded solution trajectories. With this emphasis, in this section only the concept of boundedness is introduced with Definition 2.8.
Definition 2.8: [49] Uniform Boundedness The solutions of (2.3) are
• uniformly bounded if there exists a positive constant c, inde- pendent of t0> 0, and for every a∈ (0, c), there is β = β(a) > 0, independent of t0, such that
kx(t0)k 6 a ⇒ kx(t)k 6 β, ∀t > t0
• globally uniformly bounded if (2.2.2) holds for arbitrarily large a.
• uniformly ultimately bounded with ultimate bound b if there exist positive constants b and c, independent of t0 > 0 , and for every a∈ (0, c), there is T = T (a, b) > 0, independent of t0, such that
kx(t0)k 6 a ⇒ kx(t)k 6 b, ∀t > t0+ T
• globally uniformly ultimately bounded if (2.2) holds for arbitrarily large a.
2.3 Adaptive Control
The term adaptive control is in literature generally used as a rather ambiguous term. From the application of the first adaptive control until today it is inter- preted in many different ways. In K.J. ˚Astr¨om’s Adaptive Control, a general definition is provided with the following statement.
“An adaptive controller is a controller with adjustable parameters and a mechanism for adjusting the parameters.”
Karl Johan ˚Astr¨om [54]
This definition indicates the broad range of methods that fall under the term adaptive control.
Essentially, adaptive control systems consider some model that is assumed to deviate from the real system by introducing some additional assumptions. These deviation assumptions are then aimed to be counteracted using adaptive control methods.
For differentiating various kind of methods, these are categorised with the fol-
lowing terms for a system generally described by ξ = f (t, ξ, u, δ),˙
where ξ describe the system states, f the system dynamics, u the control inputs and δ the disturbance acting on this system. Consequentially, a model is derived yielding
ξ = ¯˙ f (t, ξ, u, ¯δ),
with some model ¯f of the system dynamics and some ¯δ defining assumptions about the uncertainty.
Now with the model ¯f an adaptive control system is derived with the time varying control u = uad(t) and categorised using the following terms for system dynamics, estimation methods and disturbance characteristics.
• System dynamics – Nonlinear:
∗ General class: ¯f = ¯f (ξ, uad) + ¯δ = ¯f (ξ) + ¯fu(ξ, uad) + ¯δ
∗ Input affine: ¯f = ¯fξ(ξ) + ¯fu(ξ)uad+ ¯δ
– Linear5: ¯f = ¯f (ξ) + ¯fu(ξ, uad) + ¯δ = Aξ + Buad+ ¯δ
• Estimation method
– Direct: uad= gad(ki(t))
(control law based on parameters which are found by algebraic rela- tion)
– Indirect: uad= gad(ki(t, ˆpi))
(control law based on estimation of parameters which are found as a result of defining differential relations)
• Disturbance characteristic – Structure
∗ Structured: ¯δ = wTξ
(parametric uncertainty: known analytic expression with un- known parameters)
∗ Unstructured: ¯δ = ¯fδ(ξ, u) (unknown analytic expression) – Control authority
∗ Matched: ¯δ ∈ U
(can be matched by the control input)
∗ Unmatched: ¯δ /∈ U
(cannot be matched by the control input)
where (¯·) indicates the modelling assumptions, kiwith i∈ Z indicating multiple control parameters andU being the image of the control function fu: Rdu→ U for control input u∈ Rdu. A more descriptive introduction of these categorisa- tion is provided in [51, p. 86].
The following sections are loosely based on [54,53,51].
So from a general perspective adaptive control searches a time-varying control law u(t) that achieves desired plant characteristics. In most cases these charac- teristics are defined by the system poles, which then yields control laws defined to achieve a system configuration representing these poles. This method is called adaptive pole placement control (APPC).
A variation of this method is model reference adaptive control [53].
5 So that ¯f (ξ) = Aξ and ¯fu(ξ, uad) = Buad
In this method the control law is designed such that the resulting system con- verges to some designer-chosen reference trajectory. Here the reference tra- jectory incorporates the performance expectations of the plant and is usually generated by a reference model. The overall architecture is depicted in Figure 2.2 with employing approximate model inversion (AMI) [39].
◦ ξ˙rm= frm(ξrm, r) + f¯−1(ϑ, ξ) ξ = f (ξ, µ) + δ˙ −1 +
ϑ =−Θ(t)TΦ(t, ξ) Θ = f˙ Θ(t, ξ, e)
ϑpd= fpd(e) +
Reference Model Approximate Model Inversion
Plant
Parameter Adjustment Mechanism
Error Feedback Controller r(t)
ξrm
ξ˙rm ϑ µ ξ
e ϑad
ϑpd
Figure 2.2: Overview of a model reference adaptive control architecture using approximate model inversion.
2.4 Gaussian Processes
As a natural extension of adaptive control and its preconfigured adjustment mechanism for specific scenarios, learning mechanisms arise as an adjustment mechanism for scenarios that have not been previously encountered. With high performance computer systems and optimised algorithms, statistical models can be employed to unveil patterns and apply inference on unseen data. The study of such methods is considered as machine learning.
In the context of this thesis machine learning is applied to investigate the es- timation of a dynamical multidimensional function f with the input/output relation
f (X ) = Y,
where the input is X ∈ Rmand the output is Y ∈ Rn.
This means that the relationship between dynamical inputs X (t) and dynamical outputs Y(t) is investigated considering methods from machine learning. If
hereby both X and Y are available and utilised in the estimation process and furthermore the output domain for Y is real valued, the problem is labeled a regression problem and we speak of supervised learning.
As the approximation of this nonlinear function is an estimation process, it is natural to consider probabilities for the modelling of uncertainty in this estima- tion process.
2.4.1 Gaussian Processes
In this thesis the dynamical evolution of a mechanical system is investigated.
From experience in classical control theory it is known that uncertainty in model- basedcontrol originates commonly from model mismatch in the form of param- eter deviations, unmodelled dynamics or external disturbances. As a result, the expected uncertainty to be modelled is assumed to be epistemic, rather than purely aleatory.
This assumption of some regularity, respectively structure in the uncertainty, constitutes the basis for applying a Bayesian methodology. Where the concept of Bayesian inference treats uncertainty in a subjective way considering recent evidence for inference. Which is contrary to a frequentist approach, that is dominated by the focus on objectivity.
The elementary method in Bayesian inference is founded in Bayes’ theorem6, which is stated in Theorem 2.3.
Theorem 2.3: Bayes’ Theorem
P (A|B) = P (B|A)P (A) P (B) ,
with P (A|B) being the conditional probability of A given B, P (B|A) the conditional probability of B given A, P (A) and P (B) marginal like- lihoods, defined by P (B) =R
AP (B|A)P (A).
With this theorem a method for propagating probabilities is provided and allows to update a prior distribution in case of new evidence.
With the estimation of a continuous nonlinear dynamical function the notion of similarity comes to mind. In specific this means that similar inputs should produce similar outputs. A common method that encodes similarity and pe- riodicity in its input to output estimation, is a Gaussian process that in fact models its data points as jointly Gaussian. Following this method, the multi- dimensional nonlinear dynamical function f is modelled by a Gaussian process such that
6Also called Bayes’ rule.
