DE102022206381A1 - Method for controlling a robotic device - Google Patents

Abstract

According to various embodiments, a method for controlling a robotic device is described, comprising providing demonstrations of a robotic capability, each demonstration showing a trajectory comprising a sequence of robot configurations, each robot configuration being at least partially described by a point on a first Riemannian manifold, Determining a vector field on a second Riemann manifold and training a diffeomorphism between the first Riemann manifold and the second Riemann manifold. The method further includes controlling the robotic device, for a configuration for which a control action is to be determined, according to a configuration change given by the determined vector field via the trained diffeomorphism for the configuration for which a control action is to be determined.

Description

State of the art

The present invention relates to methods for controlling a robotic device.

In many applications, it is desirable that robots can function autonomously in potentially dynamic and unstructured environments. To do this, they have to learn how to move and interact with their environment. To do this, robots may rely on a library of skills that can be used to perform simple movements or to perform complicated tasks such as composing multiple skills. One way to learn movement skills is through human examples, known as Learning from Demonstrations (LfD). This involves a (typically human) expert demonstrating once or several times a specific skill (e.g. a movement) that is intended to be imitated by a robot.

Learning from demonstrations allows robots to directly acquire human-like abilities to perform a wide variety of tasks. Under the LfD framework, demonstrations are usually represented as state-action trajectories, for which a state can be, for example, the robot end-effector position, and an action can be the required Cartesian velocities. In the context of robot motion learning, approaches such as behavioral cloning offer to learn a direct mapping (i.e., a function) from states to actions using demonstration data. However, these methods can suffer from data misfitting and extrapolation (generalization) difficulties, sometimes resulting in unreliable motions when generated outside of demonstration data support. One way to overcome this is to leverage control theory, particularly stability certificates, which can improve the generalization capabilities of motion learning methods by introducing stability guarantees. Therefore, the problem of robot motion learning requires building a model that encapsulates the demonstrated motion dynamics and guarantees stability.

In these aspects, the usual assumption is that the robot motion trajectories lie in Euclidean spaces. However, this assumption does not strictly hold when considering the movement of the robot end effector due to the orientation representation. For example, there are different types of representation for orientation, such as quaternions in the 3-sphere manifold and rotation matrices in the special orthogonal group (SO(3)) manifold.

Therefore, approaches for robust training of a robotic device, particularly for learning from demonstrations, that can effectively handle the geometric constraints of the demonstration data in non-Euclidean spaces are desirable.

Disclosure of the invention

According to various embodiments, a method of controlling a robotic device is provided, including: providing demonstrations of a robotic capability, each demonstration showing a trajectory that includes a sequence of robot configurations, each robot configuration at least partially described by a point on a first Riemannian manifold is, determining a vector field on a second Riemann manifold and training a diffeomorphism between the first Riemann manifold and the second Riemann manifold by searching for parameter values of the diffeomorphism that minimize loss. The loss includes, for each demonstration and for at least one robot configuration of the trajectory shown by the demonstration, a difference between a configuration change to the next configuration of the trajectory and a configuration change given by the particular vector field via the diffeomorphism. Determining the configuration change given by the particular vector field for a configuration involves mapping the point on the first Riemann manifold describing the configuration to a point on the second Riemann manifold using the diffeomorphism, determining a change given by the vector field at the point on the second Riemann manifold is specified, and mapping the determined change to the tangent space of the point on the first Riemann manifold using the differential of the inverse diffeomorphism, where the configuration change is given by the result of the mapping. The method further includes controlling the robot device, for a configuration for which a control action is to be determined, according to a configuration change determined by the specific vector field is given via the trained diffeomorphism for the configuration for which a control action is to be determined.

The method described above allows learning from demonstrations for dynamic motion from a Riemannian perspective, where position-velocity pairs (generally pairs of configuration (state) and a configuration change (action) that should be performed in the configuration) on a vector field describe a Riemannian manifold (which corresponds to a state-action mapping function). The underlying Riemannian geometry is integrated during the learning process to ensure that the learned vector field (on the first Riemannian manifold) satisfies the underlying geometry of the demonstration data. For this purpose, a diffeomorphism is used, which, if the particular vector field (on the second Riemann manifold) is stable, guarantees the stability (e.g. in the Lyapunov sense) of the learned control strategy (given by the learned vector field on the first Riemann manifold). .

This can be done by determining a (e.g. canonical) stable vector field according to the Lyapunov stability requirements on a predetermined space (here the second Riemannian manifold, e.g. for orientations, but the same can also be done for the Euclidean space for positions, where the situation is much simpler). Then, a vector field is provided on the first Riemann manifold by learning a diffeomorphism between the desired (observed) demonstration data and the second Riemann manifold (representing a stable system by means of the (pre)determined vector field). The learned vector field can be used to reproduce stable dynamics in the control that mimic the demonstration trajectories.

The above approach can be applied to a wide variety of applications where data lies on a Riemannian manifold. It guarantees global asymptotic stability of the robotic device along with high accuracy of motion reproduction.

Various examples are given below.

Example 1 is a method of controlling a robot device as described above.

Example 2 is the procedure of Example 1, including determining that the vector field is a stable vector field.

Thus, it can be achieved that the control actions taken according to the configuration changes given by the particular vector field via the diffeomorphism lead to stable and efficient control of the robotic device.

Example 3 is the method of Example 1 or 2, including training the diffeomorphism by training a neural network, the diffeomorphism mapping a point to the solution of a neural ordinary differential equation, where the value of the derivative at a point is given by the output of the neural network, when it is supplied with the point, is given.

Thus, diffeomorphism can be trained efficiently by training a neural network since efficient training methods are available for neural networks. Furthermore, a neural ordinary differential equation can be solved efficiently in a forward iteration.

Example 4 is the method of any of Examples 1 to 3, including determining the vector field on the second Riemann manifold depending on the output of another neural network and training the further neural network together with the diffeomorphism by searching for parameter values of the diffeomorphism and the further neural network to minimize loss.

Thus, additional degrees of freedom are introduced in the form of the additional neural network, which enables better adaptation to the demonstrations. The further neural network, for example, receives points on the second Riemann manifold as input.

Example 5 is the method of any of Examples 1 to 4, wherein each robot configuration includes an orientation of a component of the robot device described by the respective point on the first Riemann manifold.

Orientations can be described more efficiently using (general) Riemannian manifolds rather than Euclidean spaces.

Example 6 is a robot device controller designed to carry out the method of any of Examples 1 to 5.

Example 7 is a computer program including instructions that, when executed by a processor, cause the processor to perform a method according to any of Examples 1 to 5.

Example 8 is a computer-readable medium that stores instructions that, when executed by a processor, cause the processor to perform a method according to any of Examples 1 to 5.

• 1 zeigt einen Roboter.
• 2 zeigt eine Veranschaulichung einer Sphären-Mannigfaltigkeit S², deren Punkte zum Beispiel jeweils eine mögliche Orientierung eines Roboterendeffektors repräsentieren können.
• 3 veranschaulicht einen Diffeomorphismus zwischen einer ersten riemannschen Mannigfaltigkeit und einer zweiten riemannschen Mannigfaltigkeit, der zum Ableiten vom Robotersteueraktionen verwendet wird, gemäß verschiedenen Ausführungsformen.
• 4 veranschaulicht eine numerische Integration auf einer Mannigfaltigkeit unter Verwendung eines Projektionsverfahrens.
• 5 zeigt ein Flussdiagramm, das ein Verfahren zum Steuern einer Robotervorrichtung veranschaulicht, gemäß einer Ausführungsform.

In the drawings, similar reference numerals generally indicate the same parts throughout the different views. The drawings are not necessarily to scale, but rather emphasis is generally placed on illustrating the principles of the invention. In the following description various aspects will be described with reference to the following drawings, in which:

• 1 shows a robot.
• 2 shows an illustration of a sphere manifold S ² , the points of which can each represent, for example, a possible orientation of a robot end effector.
• 3 illustrates a diffeomorphism between a first Riemannian manifold and a second Riemannian manifold used to derive robot control actions, according to various embodiments.
• 4 illustrates numerical integration on a manifold using a projection method.
• 5 shows a flowchart illustrating a method for controlling a robotic device, according to an embodiment.

The following detailed description refers to the accompanying drawings, which illustratively show specific details and aspects of this disclosure in which the invention may be practiced. Other aspects may be utilized and structural, logical and electrical changes may be made without departing from the scope of the invention. The various aspects of this disclosure are not necessarily mutually exclusive, as some aspects of this disclosure may be combined with one or more other aspects of this disclosure to form new aspects.

Various examples are described in more detail below.

1 shows a robot 100.

The robot 100 includes a robot arm 101, for example an industrial robot arm, for handling or assembling a workpiece (or one or more other objects). The robot arm 101 includes manipulators 102, 103, 104 and a base (or support) 105 with which the manipulators 102, 103, 104 are supported. The term “manipulator” refers to the movable links of the robot arm 101, the actuation of which enables physical interaction with the environment, e.g. B. to carry out a task. For control, the robot 100 includes a (robot) controller 106, which is designed to implement interaction with the environment according to a control program. The last member 104 (farthest from the support 105) of the manipulators 102, 103, 104 is also referred to as the end effector 104 and may include one or more tools such as a welding torch, a gripping instrument, a painting device or the like.

The other manipulators 102, 103 (closer to the support 105) can form a positioning device so that, together with the end effector 104, the robot arm 101 is provided with the end effector 104 at its end. The robot arm 101 is a mechanical arm that can provide similar functions to a human arm (possibly with a tool at the end).

The robot arm 101 may include joint elements 107, 108, 109 that connect the manipulators 102, 103, 104 to each other and to the support 105. A joint element 107, 108, 109 can have one or more Have joints that can each provide a rotatable movement (ie, rotational movement) and/or a translational movement (ie, displacement) for associated manipulators relative to one another. The movement of the manipulators 102, 103, 104 can be initiated by means of actuators that are controlled by the controller 106.

The term “actuator” can be understood as a component that is designed to influence a mechanism or process in response to a control. The actuator can implement instructions created by the controller 106 (the so-called activation) into mechanical movements. The actor, e.g. B. an electromechanical transducer can be designed to convert electrical energy into mechanical energy in response to a control.

The term “controller” can be understood as any type of logic-implementing entity, which may include, for example, a circuit and/or a processor capable of controlling software stored in a storage medium, firmware, or a combination of it, and can create the instructions, e.g. B. to an actuator in the present example. The controller may be configured, for example, through program code (e.g., software) to control the operation of a system, a robot in the present example.

In the present example, the controller 106 includes one or more processors 110 and a memory 111 that stores code and data based on which the processor 110 controls the robot arm 101. According to various embodiments, the controller 106 controls the robot arm 101 based on a machine learning model 112 stored in the memory 111.

aus Datenpunkten ${\text{y}}_t\in {ℝ}^3\times S^3,$

d. h. Paaren von Punkten, wobei der erste Punkt (von ℝ³) zum Beispiel die Endeffektorposition spezifiziert und der zweite Punkt eine Quaternion (von

) ist und die Orientierung des Endeffektors spezifiziert. Quaternionen weisen vorteilhafte Eigenschaften für die Robotersteuerung auf. Da Quaternionen (die für die Robotersteuerung verwendet werden) jedoch eine Einheitsnorm-Beschränkung erfüllen, bilden sie keinen Vektorraum. Sie liegen jedoch auf der riemannschen Mannigfaltigkeit

Ein anderes Beispiel für eine riemannschen Mannigfaltigkeit, deren Elemente (d. h. Punkte) für die Robotersteuerung verwendet werden können, ist die spezielle orthogonale Gruppe (SO(3)).The machine learning model 112 is trained by learning from demonstrations, e.g. B. be provided by a user who moves the robot arm 101 manually. Each demonstration is a sequence of robot configurations. According to various embodiments, each configuration is at least partially given by a point on a Riemannian manifold. For example, there is a certain demonstration trajectory $\mathrm{\tau }={\left\{{\text{y}}_t\right\}}_{t=1}^T$

from data points ${\text{y}}_t\in {ℝ}^3\times S^3,$

i.e. pairs of points, for example where the first point (of ℝ ³ ) specifies the end effector position and the second point a quaternion (of

) and the orientation of the end effector is specified. Quaternions have advantageous properties for robot control. However, since quaternions (used for robot control) satisfy a unit norm constraint, they do not form a vector space. However, they lie on the Riemannian manifold

Another example of a Riemannian manifold whose elements (i.e. points) can be used for robot control is the special orthogonal group (SO(3)).

ist ein m-dimensionaler topologischer Raum, für den jeder Punkt einem euklidischen Raum ℝ^m gleicht und der eine global definierte differenzielle Struktur aufweist. Für jeden Punkt $x\in M$

existiert ein Tangentialraum $T_{x}M$

der ein Vektorraum ist, der aus den Tangentialvektoren aller möglichen glatten Kurven, die durch x laufen, besteht. Eine riemannsche Mannigfaltigkeit ist mit einem gleichmäßig variierenden positiven definitiven inneren Produkt genannt eine riemannsche Metrik versehen, die gestattet, Kurvenlängen in

zu definieren. Diese Kurven, Geodäten genannt, sind die Generalisierung gerader Linien im euklidischen Raum auf riemannsche Mannigfaltigkeiten, da sie die Kurven mit minimaler Länge zwischen zwei Punkten in

repräsentieren.A Riemannian manifold

is an m-dimensional topological space for which every point is equal to a Euclidean space ℝ ^m and which has a globally defined differential structure. For every point $x\in M$

there is a tangent space $T_{x}M$

which is a vector space consisting of the tangential vectors of all possible smooth curves passing through x. A Riemannian manifold is provided with a uniformly varying positive definite inner product called a Riemannian metric, which allows curve lengths in

define. These curves, called geodesics, are the generalization of straight lines in Euclidean space to Riemannian manifolds, since they represent the minimum length curves between two points in

represent.

deren Punkte zum Beispiel jeweils eine mögliche Orientierung eines Roboterendeffektors repräsentieren können. 2 shows an illustration of a sphere manifold

whose points can each represent, for example, a possible orientation of a robot end effector.

Two points x and y are indicated on the sphere that can be used by the controller 106 to represent two different orientations of the robot end effector 104.

The shortest distance between the two points in the surrounding space would be a straight line 201, while the shortest path on the manifold is a geodesic 202.

verwendet werden, die als die Exponentialabbildung bzw. die Logarithmusabbildung bezeichnet werden.To use the Euclidean tangent spaces, mappings can be made back and forth between the tangent spaces $T_{x}M$

are used, which are referred to as the exponential map and the logarithm map, respectively.

bildet einen Punkt u im Tangentialraum von x auf einen Punkt y auf der Mannigfaltigkeit ab, sodass er auf der Geodäte beginnend bei x in die Richtung u liegt, sodass die geodätische Distanz zwischen x und y gleich der Norm der Distanz zwischen x und u ist. Die inverse Operation wird die Logarithmusabbildung ${\text{Log}}_x\text{:}M\to T_{x}M$

genannt, d. h. u = Log_x(y).The exponential map ${\text{Exp}}_x\text{:}{T}_{x}M$

maps a point u in the tangent space of x to a point y on the manifold so that it lies on the geodesic starting at x in the direction u, so that the geodesic distance between x and y is equal to the norm of the distance between x and u. The inverse operation becomes the logarithm map ${\text{log}}_x\text{:}M\to T_{x}M$

called, i.e. u = log _x (y).

der Elemente zwischen Tangentialräumen verschiebt, sodass das innere Produkt zwischen zwei Elementen im Tangentialraum konstant bleibt.Another useful operation over manifolds is parallel transport ${\mathrm{\Gamma }}_{x\to y}\text{:}{T}_{x}M\to T_{y}M,$

of the elements between tangent spaces, so that the inner product between two elements in the tangent space remains constant.

nach $T_{y}M$

(der Index x→y wird der Einfachheit halber weggelassen).For example in 2 Γ(v ₁ ), Γ (v ₂ ) are the parallel transported vectors v ₁ and v ₂ of $T_{x}M$

after $T_{y}M$

(the index x→y is omitted for simplicity).

Operation Formel $d_M\left(\text{x},\text{y}\right)$

arccos(x^Ty) Exp_x(u) $\text{xcos}\left(\Vert \text{u}\Vert \right)+\overline{\text{u}}\text{sin}\left(\Vert \text{u}\Vert \right)\text{mit }\overline{\text{u}}=\frac{\text{u}}{\Vert \text{u}\Vert }$

Log_x(y) $d_M\left(\text{x}\text{,y}\right)\frac{y-{\text{x}}^{\text{T}}\text{y x}}{\Vert \text{y}-{\text{x}}^{\text{T}}\text{y x}\Vert }$

Γ_x→y(v) $\begin{array}{l}\left(-\text{xsin}\left(\Vert \text{u}\Vert \right){\overline{\text{u}}}^{\text{T}}+\overline{\text{u}}\text{cos}\left(\Vert \text{u}\Vert \right){\overline{\text{u}}}^{\text{T}}+\left(\text{I}-{\overline{\text{u}}\overline{\text{u}}}^{\text{T}}\right)\right)\text{v mit }\overline{\text{u}}=\\ \frac{\text{u}}{\Vert \text{u}\Vert }\text{und u}={\text{Log}}_{\text{x}}\left(\text{y}\right)\end{array}$

The following are the expressions for the Riemann distance, exponential and logarithmic maps, and parallel transport operation for the sphere manifold

surgery formula $d_M\left(\text{x},\text{y}\right)$

arccos(x ^T y) Exp _x (u) $\text{xcos}\left(\Vert \text{u}\Vert \right)+\overline{\text{u}}\text{sin}\left(\Vert \text{u}\Vert \right)\text{with}\overline{\text{u}}=\frac{\text{u}}{\Vert \text{u}\Vert }$

Log _x (y) $d_M\left(\text{x}\text{,y}\right)\frac{y-{\text{x}}^{\text{T}}\text{yx}}{\Vert \text{y}-{\text{x}}^{\text{T}}\text{yx}\Vert }$

Γ _x→y (v) $\begin{array}{l}\left(-\text{xsin}\left(\Vert \text{u}\Vert \right){\overline{\text{u}}}^{\text{T}}+\overline{\text{u}}\text{cos}\left(\Vert \text{u}\Vert \right){\overline{\text{u}}}^{\text{T}}+\left(\text{I}-{\overline{\text{u}}\overline{\text{u}}}^{\text{T}}\right)\right)\text{v with}\overline{\text{u}}=\\ \frac{\text{u}}{\Vert \text{u}\Vert }\text{and u}={\text{log}}_{\text{x}}\left(\text{y}\right)\end{array}$

The task of learning from demonstrations can be viewed as learning a vector field in the robot's respective space of configurations. In the following the space of configurations is a Riemannian manifold. However, it should be noted that any configuration may have additional components, in particular a component in Euclidean space (like ℝ ³ in the example above), which is omitted below for simplicity (and which is easier to treat than the more general case of Riemannian diversity).

Thus, an approach for learning a vector field on a Riemannian manifold while guaranteeing global asymptotic stability is described below. According to various embodiments, this is accomplished by utilizing a diffeomorphism that transforms a target vector field (i.e., the one that should be learned) on the Riemann manifold to a simple canonical (i.e., predetermined) stable vector field on a latent manifold. The diffeomorphism is implemented by a (first) neural network, as described in more detail below.

3 illustrates a diffeomorphism Φ _θ between a first Riemannian manifold 301 (the one on which a vector field is to be learned, i.e. whose points (at least partially) specify robot configurations) and a second Riemannian manifold 302 (i.e. the one referred to above as the Riemannian manifold) . On the first Riemannian manifold 301, a target vector field (represented here by the S-shape) should be provided (i.e. learned from the demonstrations). On the second Riemannian manifold 302, a vector field is predefined (represented by the “meridians”). It should be noted that the Riemann manifolds 301, 302 are the same in a topological sense, i.e. there is a homeomorphism between the two manifolds.

For example, a C1 diffeomorphism is learned using continuous normalization flows on manifolds. Before learning the diffeomorphism, the target vector field 302 on the latent manifold 302 is selected. As expected, one of the conditions that must be satisfied when selecting a latent manifold is that the latent manifold is 302 and the original manifold (i.e. the first manifold 301) must be diffeomorphic. This target vector field 302 is predefined to behave as a geodesic vector field, with the path integral on this vector field producing geodesics on the latent manifold.

kann der Tangentialvektor ẏ am Ort $y\in N$

berechnet werden als $$\dot{y}=g\left(y\right)=k_{\gamma }\left(y\right)\frac{Lo{g}_y\left(y_0\right)}{{\Vert Lo{g}_y\left(y_0\right)\Vert }_2}.$$

To construct a stable target vector field 302, the logarithm map can be used to cause the vector field to follow the geodesics. Especially considering an attractor point $y_0\in N$

can the tangential vector ẏ at the location $y\in N$

be calculated as $$\dot{y}=G\left(y\right)=k_{\gamma }\left(y\right)\frac{LO{G}_y\left(y_0\right)}{{\Vert LO{G}_y\left(y_0\right)\Vert }_2}.$$

d. h. $k_{\gamma }\left(y\right)=e^{{\psi }_{\gamma }\left(y\right)+\epsilon },$

wobei ε eine sehr kleine Konstante ist (z. B. die Maschinenpräzision).According to various embodiments, the factor k _γ is constructed and trained using a (second) neural network with parameter γ to learn using the demonstration data together with the diffeomorphism Φ _θ . It should be noted that the factor k _γ only needs to be greater than zero at all locations y ≠ y ₀ for the vector field to be stable on the manifold. According to various embodiments, the exponential function is used, the exponent of which is modeled by the second neural network ${\psi }_{\gamma }:{ℝ}^n\supset N\to ℝ,$

ie $k_{\gamma }\left(y\right)=e^{{\psi }_{\gamma }\left(y\right)+\epsilon },$

where ε is a very small constant (e.g. the machine precision).

bezeichnet werden und das Zielvektorfeld soll mit $V\in T\mathcal{M}$

bezeichnet werden. Das Lernen dieses Vektorfeldes bedeutet das Ermöglichen der Evaluierung der Geschwindigkeit vx am Ort $x\in \mathcal{M}.$

Es sollte angemerkt werden, dass ein Vektorfeld als eine Abbildung zwischen der Mannigfaltigkeit und ihrem Tangentialraum $v:\mathcal{M}\to T\mathcal{M}$

angesehen werden kann.The first manifold 301 should be with

and the target vector field should be denoted by $v\in T\mathcal{M}$

be referred to. Learning this vector field means enabling the evaluation of the velocity vx at the location $x\in \mathcal{M}.$

It should be noted that a vector field is considered a mapping between the manifold and its tangent space $v:\mathcal{M}\to T\mathcal{M}$

can be viewed.

die angewendet werden soll, um sie auf die latente Mannigfaltigkeit 302 zu projizieren, bezeichnet mit

gelernt.According to various embodiments, instead of directly parameterizing the target vector field, the diffeomorphic mapping ${\mathrm{\Phi }}_{\theta }:\mathcal{M}\to N,$

which is to be applied to project onto the latent manifold 302, denoted by

learned.

gegeben. Aufgrund der Tatsache, dass ein Diffeomorphismus topologische Eigenschaften bewahrt, kann erwartet werden, dass das auf diese Weise gelernte Zielvektorfeld weiterhin global stabil ist, solange das geodätische Vektorfeld auf der latenten Mannigfaltigkeit stabil ist.Learning the target vector field thus involves learning the diffeomorphism (i.e. adjusting its parameters θ, which are, for example, weights of a neural network to implement the diffeomorphism, as described in more detail below) so that it matches the target vector field to the predefined (globally stable) vector field on the second manifold 302 is projected. The second manifold is, for example, by a geodesic vector field $G_{\gamma :N\to TN}$

given. Due to the fact that a diffeomorphism preserves topological properties, the target vector field learned in this way can be expected to continue to be globally stable as long as the geodesic vector field on the latent manifold is stable.

$$

spezifiziert wird, durch

• • Abbilden des Punktes x auf die latente Mannigfaltigkeit 302 mit dem gelernten Diffeomorphismus ${\Phi }_{\theta }:\mathcal{M}\to N,$
  
• • Evaluieren des geodätischen Vektorfeldes auf die bestimmte Änderung (z. B. Geschwindigkeit) für die latente Variable y (d. h. den Punkt auf der zweiten riemannschen Mannigfaltigkeit, auf den x mittels des Diffeomorphismus abgebildet wurde)
• • Verwenden des Pullback-Operators $D_y{\mathrm{\Phi }}_{\theta }^{*},$
  
  um die bestimmte Änderung zurück auf den Tangentialraum der ersten Mannigfaltigkeit 301 am Ort $x\in \mathcal{M}$
  
  zu projizieren. Der Pullback-Operator kann als die inverse Abbildung des Differentials des Diffeomorphismus angesehen werden, d. h. die Inverse von $D_x{\mathrm{\Phi }}_{\theta }:T_{{\mathrm{\Phi }}_{\theta }\left(x\right)}N\to T_{x}N,$
  
  die durch Berechnen der Ableitungen von Φ_θ und dann Ableiten der Inversen oder Berechnen der Ableitungen des inversen Diffeomorphismus bestimmt werden kann.

After learning the diffeomorphism and its inverses, to determine a control action for a current robot configuration, the respective controller (e.g., controller 106) can determine the action for a configuration (ie, the configuration change such as speed) that passes through the point $x\in \mathcal{M}$

$$

is specified by

• • Mapping the point x onto the latent manifold 302 using the learned diffeomorphism ${\Phi }_{\theta }:\mathcal{M}\to N,$
  
• • Evaluate the geodesic vector field for the specific change (e.g. velocity) for the latent variable y (i.e. the point on the second Riemann manifold to which x was mapped using diffeomorphism)
• • Using the pullback operator $D_y{\mathrm{\Phi }}_{\theta }^{*},$
  
  around the specific change back to the tangent space of the first manifold 301 at the location $x\in \mathcal{M}$
  
  to project. The pullback operator can be viewed as the inverse map of the differential of the diffeomorphism, that is, the inverse of $D_x{\mathrm{\Phi }}_{\theta }:T_{{\mathrm{\Phi }}_{\theta }\left(x\right)}N\to T_{x}N,$
  
  which can be determined by computing the derivatives of _Φ and then deriving the inverses or computing the derivatives of the inverse diffeomorphism.

As in 3 illustrated, this procedure can be combined into a single equation as follows $$\dot{x}=\left(D_y{\mathrm{\Phi }}_{\theta }^{*}\circ G_{\gamma }\circ {\Phi }_{\theta }\right)\left(x\right)=D{\Phi }_{\theta }^{*}\left(\dot{y}\right)$$

As mentioned above, continuous normalization flows on manifolds can be used to construct the (trainable) diffeomorphism.

wobei f_θ ein vorwärtsgekoppeltes neuronales Netzwerk bezeichnet und z_k einen k-ten verborgenen Zustand repräsentiert.In particular, according to various embodiments, a residual flow is used to construct the diffeomorphism. A residual flow can be represented as the composition of residual connections of the following form $${\mathrm{e.g}}_{k+1}={\mathrm{e.g}}_k+f_{\theta }\left({\mathrm{e.g}}_k\right)$$

where f _θ denotes a feedforward neural network and z _k represents a kth hidden state.

Iterating through equation (2) (from a starting index s to an ending index e) results in computing the transformation of a point z _s = x to a point y = z _e . Furthermore, by limiting the Lipschitz constant of f _θ to less than 1, ie Lip(f _θ ) < 1, the invertibility of the residual blocks can be sufficiently guaranteed.

The residual flow can be viewed as an Euler discretization of a continuous ordinary differential equation (ODE) because the neural network f _θ is called a neural ODE, $$\frac{d}{dt}\mathrm{e.g}\left(t\right)=f_{\theta }(\mathrm{e.g}\left(t\right),t)$$

wobei das Ergebnis der Berechnung auf der rechten Seite als ein Punkt der zweiten Mannigfaltigkeit 302 mittels des Homöomorphismus zwischen den beiden Mannigfaltigkeiten angesehen wird.The diffeomorphic map given by the residual flow (and thus given by the neural network and its weights) can be calculated by iteratively traversing the residual network f _θ . This means that computing the diffeomorphism is equivalent to solving an initial value problem (or integral). To do this, the starting point x = z(t _s ) ∈ ℝ ⁿ is used to calculate y by $$y={\Phi }_{\theta }\left(x\right)=x+{\displaystyle {\int }_{\tau =t_s}^{t_e}{f}_{\theta }(\mathrm{e.g}\left(\tau \right)},\tau )d\tau$$

where the result of the calculation on the right is viewed as a point of the second manifold 302 by means of the homeomorphism between the two manifolds.

kann leicht durch Umdrehen des Zeitintervalls für die Integration erhalten werden, d. h. $$x={\Phi }_{\theta }^{-1}\left(\text{y}\right)=\text{y}+{\displaystyle {\int }_{\tau =t_e}^{t_s}{f}_{\theta }(z\left(\tau \right)},\tau )d\tau$$

(z. B. entsprechend dem maschinellen Lernmodell 112)The inverse diffeomorphism ${\mathrm{\Phi }}_{\theta }^{-1}$

can be easily obtained by flipping the time interval for integration, ie $$x={\Phi }_{\theta }^{-1}\left(\text{y}\right)=\text{y}+{\displaystyle {\int }_{\tau =t_e}^{t_s}{f}_{\theta }(\mathrm{e.g}\left(\tau \right)},\tau )d\tau$$

(e.g. according to machine learning model 112)

und $f_{\theta }:\mathcal{M}\times ℝ\to T\mathcal{M}.$

Dies führt zu der Formulierung neuronaler Mannigfaltigkeits-ODEs. Es sollte angemerkt werden, dass neuronale ODEs lediglich ein Spezialfall neuronaler Mannigfaltigkeits-ODEs sind, wenn die Mannigfaltigkeit euklidisch ist. Da der euklidische Raum lediglich ein Spezialfall von Mannigfaltigkeit ist, sind alle Verweise auf Gleichung (3) im Folgenden im Zusammenhang mit neuronalen Mannigfaltigkeits-ODE gemeint.The above describes constructing a diffeomorphism under the Euclidean setting. To account for the underlying geometry of data (i.e. configurations that are at least partially represented by points on a Riemannian manifold), the continuous normalization flow is extended to a manifold environment, i.e. where $x,{\mathrm{e.g}}_k\in \mathcal{M}$

and $f_{\theta }:\mathcal{M}\times ℝ\to T\mathcal{M}.$

This leads to the formulation of neural manifold ODEs. It should be noted that neural ODEs are merely a special case of neural manifold ODEs when the manifold is Euclidean. Since Euclidean space is merely a special case of manifold, all references to equation (3) below are meant in the context of neural manifold ODE.

Furthermore, in the general manifold case, the diffeomorphism can be calculated using equation (4) by performing the integration over the manifold rather than in Euclidean space.

Integration on manifolds can be performed using projection methods and integrators based on local coordinates.

4 illustrates numerical integration on a manifold using a projection method.

wobei $f_{\theta }\left(\cdot ,t\right)\in T_{z\left(t\right)}\mathcal{M}$

für alle $z\left(t\right)\in \mathcal{M},$

berechnet das Projektionsverfahren ein Ein-Schritt-Integral mit einem willkürlichen numerischen Integrator im euklidischen Raum und projiziert dann den Wert auf die Mannigfaltigkeit

um die Integralergebnisse auf der Mannigfaltigkeit zu erhalten, wie in 4 veranschaulicht. Beispielsweise wird ein Euler-Löser mit Schrittgröße dt verwendet, der einen einzelnen Schritt vom Anfangspunkt z_a berechnet. Der nächste Zustand z̃₁ wird dann als z̃₁ = z₀ + f_θ (z₀, 0)dt berechnet. Dies wird bis zum Endpunkt z̃e fortgesetzt, der zurück auf die Mannigfaltigkeit projiziert wird.With an ODE given by equation (3) on a manifold

where $f_{\theta }\left(\cdot ,t\right)\in T_{\mathrm{e.g}\left(t\right)}\mathcal{M}$

for all $\mathrm{e.g}\left(t\right)\in \mathcal{M},$

The projection method calculates a one-step integral with an arbitrary numerical integrator in Euclidean space and then projects the value onto the manifold

to get the integral results on the manifold, as in 4 illustrated. For example, an Euler solver with step size dt is used, which calculates a single step from the starting point z _a . The next state z̃ ₁ is then calculated as z̃ ₁ = z ₀ + f _θ (z ₀ , 0)dt. This continues until the endpoint z̃e, which is projected back onto the manifold.

Integration methods on manifolds based on the local coordinates consider a local representation of the manifold through a coordinate map, which enables a formulation of an equivalent ODE in Euclidean space that can be solved and, using the coordinate map with inverse map, gives a solution on the manifold.

To include a manifold ODE in a deep learning framework (for training the neural network f _θ ), the gradient of the loss with respect to the weights of the neural network must be efficiently calculated. This can be done by adjoint sensitivity methods, which allow treating the ODE solver as a black box and computing gradients backwards in time by solving a second augmented adjoint ODE, rather than directly differentiating by ODE solvers. This approach is memory efficient, especially for adaptive ODE solvers that adjust the step size on the fly.

mit Dimension d spezifiziert sein, die in einem umgebenden n-dimensionalen euklidischen Raum eingebettet ist. Angesichts eines neuronalen Freiform-Netzwerks Ψ_θ : ℝⁿ × ℝ →ℝⁿ ist die Abbildung $f_{\theta }:{ℝ}^n\ast ℝ\to T\mathcal{M}$

dann gegeben durch f_θ (z(t),t) = proju ◯ Ψ_θ(z(t),t)It should be noted that using a neural network to model the dynamics of a manifold ODE imposes some limitations on the neural network. Because of this, according to various embodiments, a projection operator is added to the final layer of a “freeform” neural network (ie, a neural network without these restrictions) to project its output onto the tangent space at the input location. It should represent the robot configuration (at least partially) by points on a manifold

be specified with dimension d, which is embedded in a surrounding n-dimensional Euclidean space. Given a free-form neural network Ψ _θ : ℝ ⁿ × ℝ →ℝ ⁿ is the figure $f_{\theta }:{ℝ}^n\ast ℝ\to T\mathcal{M}$

then given by f _θ (z(t),t) = proju ◯ Ψ _θ (z(t),t)

Input data to the neural network is assumed to be on manifold 301. The neural network may be, for example, a fully connected neural network (FCNN) with Tanh activation function.

durch den Pullback-Operator $D_y{\mathrm{\Phi }}_{\theta }^{*}$

abgebildet.As described above, after obtaining the configuration change given for the point back to the tangent space at the location $x\in \mathcal{M}$

through the pullback operator $D_y{\mathrm{\Phi }}_{\theta }^{*}$

pictured.

This operator can be viewed as the inverse mapping of the differentials of the diffeomorphism, that is, the differentials that ẋ maps to ẏ: $$D_x{\mathrm{\Phi }}_{\theta }\left(x\right)\dot{x}=\dot{y}$$

In the Euclidean case, ẋ can be derived directly by multiplying ẏ by the inverse of D _x Φ _θ . However, in a manifold setting, computing the inverse directly is difficult, especially for a d-dimensional submanifold in an n-dimensional Euclidean space ℝ ⁿ . This is due to an inevitable loss of rank in the matrix representation for D _x Φ _θ (x).

For example, the tangent space of a 2-sphere represented in ℝ ³ also has the same dimension as the manifold itself, which implies that the operator D _x Φ _θ (x) is a mapping between between two 2-dimensional spaces. However, since the 2-sphere and its tangent space are embedded in the 3-dimensional surrounding Euclidean space, the matrix representation for D _x Φ _θ (x) is a 3 by 3 matrix. A 3-by-3 matrix for mapping between 2-dimensional spaces must be rank-deficient and thus cannot be directly inverted. It is also typically not helpful to compute the generalized inverse of this matrix, since this is equivalent to computing a minimum norm solution for ẋ that does not satisfy the properties of a tangential vector. According to various embodiments, this is addressed by introducing a constraint that allows ẋ to be imposed on the tangent space. In the case of a sphere manifold, the tangent vector ẋ must be perpendicular to the position vector x, which can be formulated as $$x^T\dot{x}=0$$

$${\left(\begin{array}{c}{D}_x{\Phi }_{\theta }\left(x\right)\\ x^T\end{array}\right)}^T\left(\begin{array}{c}{D}_x{\Phi }_{\theta }\left(x\right)\\ x^T\end{array}\right)\dot{x}={\left(\begin{array}{c}{D}_x{\Phi }_{\theta }\left(x\right)\\ x^T\end{array}\right)}^T\left(\begin{array}{c}\dot{y}\\ 0\end{array}\right)$$

$$\left(D_x{\mathrm{\Phi }}_{\theta }{\left(x\right)}^T{D}_x{\mathrm{\Phi }}_{\theta }\left(x\right)+x{x}^T\right)\dot{x}=D_x{\mathrm{\Phi }}_{\theta }{\left(x\right)}^T\dot{y}$$

Combining these two equations (6) and (7) gives a unique solution. A solution can be found by solving the linearly bounded problem either by creating a new system of linear equations or by applying a Lagrange multiplier. The following shows how to augment the matrix to create a new linear equation and calculate the pseudo-inverse. Since the augmented matrix has full rank, the pseudo-inverse produces the same result as the direct inverse. $$\left(\begin{array}{c}{D}_x{\Phi }_{\theta }\left(x\right)\\ x^T\end{array}\right)\dot{x}=\left(\begin{array}{c}\dot{y}\\ 0\end{array}\right)$$

$${\left(\begin{array}{c}{D}_x{\Phi }_{\theta }\left(x\right)\\ x^T\end{array}\right)}^T\left(\begin{array}{c}{D}_x{\Phi }_{\theta }\left(x\right)\\ x^T\end{array}\right)\dot{x}={\left(\begin{array}{c}{D}_x{\Phi }_{\theta }\left(x\right)\\ x^T\end{array}\right)}^T\left(\begin{array}{c}\dot{y}\\ 0\end{array}\right)$$

$$\left(D_x{\mathrm{\Phi }}_{\theta }{\left(x\right)}^T{D}_x{\mathrm{\Phi }}_{\theta }\left(x\right)+x{x}^T\right)\dot{x}=D_x{\mathrm{\Phi }}_{\theta }{\left(x\right)}^T\dot{y}$$

$$D_y{\Phi }_{\theta }^{*}={\left(D_x{\mathrm{\Phi }}_{\theta }{\left(x\right)}^T{D}_x{\mathrm{\Phi }}_{\theta }\left(x\right)+x{x}^T\right)}^{-1}{D}_x{\mathrm{\Phi }}_{\theta }{\left(x\right)}^T$$

This leads to the unique solution as well as the expression of the pullback operator D _x Φ _θ , formulated as $$\dot{x}=\left(D_x{\mathrm{\Phi }}_{\theta }{\left(x\right)}^T\right)D_x{\mathrm{\Phi }}_{\theta }\left(x\right)+{x{x}^T)}^{-1}{D}_x{\mathrm{\Phi }}_{\theta }{\left(x\right)}^T\dot{y}$$

$$D_y{\Phi }_{\theta }^{*}={\left(D_x{\mathrm{\Phi }}_{\theta }{\left(x\right)}^T{D}_x{\mathrm{\Phi }}_{\theta }\left(x\right)+x{x}^T\right)}^{-1}{D}_x{\mathrm{\Phi }}_{\theta }{\left(x\right)}^T$$

erhalten werden, der für Sphären spezifisch ist, da die in Gleichung (7) definierte Beschränkung nur für Sphären-Mannigfaltigkeiten gültig ist.Given the above derivation, the pullback operator $D_y{\mathrm{\Phi }}_{\theta }^{*}$

can be obtained, which is specific for spheres, since the restriction defined in equation (7) is only valid for sphere manifolds.

However, while in the sphere manifold environment the constraint is linear and easy to calculate or satisfy, in other manifolds it can be nonlinear and complex. Therefore, introducing a nonlinear condition into equation (6) requires an iterative procedure to solve a nonlinearly constrained optimization problem, which is computationally intensive.

basierend auf modifizierten adjungierten Verfahren eingeführt. Wie oben erwähnt, kann der Pullback-Operator auch als ein Differential eines inversen Diffeomorphismus $D_y\left({\mathrm{\Phi }}_{\theta }^{-1}\right)$

angesehen werden. Somit besteht ein Ansatz zum Bestimmen des Pullback-Operators darin, zunächst den inversen Diffeomorphismus zu berechnen und dann seine Ableitungen zu erhalten. In der Tat sind die Prozedur zum Berechnen des Diffeomorphismus und des inversen Diffeomorphismus für unsere Mannigfaltigkeits-ODE identisch, und nur das Zeitintervall muss invertiert werden, d. h. anstatt des Integrierens von t_s nach t_e, um den Diffeomorphismus zu erhalten, wird die Integration von t_e nach t_s durchgeführt, um den inversen Diffeomorphismus zu erhalten.Therefore, according to various embodiments, to generalize the computation for arbitrary manifolds, a way to determine the pullback operator becomes $D_y{\mathrm{\Phi }}_{\theta }^{*}$

based on modified adjoint methods introduced. As mentioned above, the pullback operator can also be used as a differential of an inverse diffeomorphism $D_y\left({\mathrm{\Phi }}_{\theta }^{-1}\right)$

be considered. Thus, one approach to determining the pullback operator is to first calculate the inverse diffeomorphism and then obtain its derivatives. In fact, the procedure for computing the diffeomorphism and the inverse diffeomorphism for our manifold ODE are identical, and only the time interval needs to be inverted, that is, instead of integrating from _ts to _te to obtain the diffeomorphism, the integration of t _e after t _s to obtain the inverse diffeomorphism.

Furthermore, an adjoint method can be used to calculate the differential of the inverse diffeomorphism.

For this purpose, A ^★ (t) is defined as an adjoint variable as D _z(t) Φ ^-1 . Algorithm 1 shows that the differential of the inverse diffeomorphism is given with a single pass of the ODE solver on the manifold, while the result for the diffeomorphism is obtained without any additional effort. Therefore, no forward computation is needed before proceeding with backward propagation.

The ManifoldODESolve function solves the manifold ODE (e.g. using a dynamic charting method). Here the main challenge lies in the calculation of the differential of the vector fields on the manifold D _z(t) f _θ (z(t),t), whereas only partial derivatives are required in the Euclidean situation. Directly computing the differential of vector fields can be avoided by several techniques, e.g. B. as described below.

φ_i = Log_zi , kann der inverse Diffeomorphismus ${\mathrm{\Phi }}_{\theta }^{-1}:y=z_k\mapsto z_0=x$

geschrieben werden als $$\begin{array}{l}{\mathrm{\Phi }}_{\theta }^{-1}=Lo{g}_{z_0}^{-1}\circ {\widehat{f}}_0{}^{-1}\circ Ex{p}_{z_0}^1\circ \dots \circ Lo{g}_{z_{k-1}}^{-1}\circ {\widehat{f}}_{k-1}{}^{-1}\circ Ex{p}_{z_{k-1}}^1\\ =Ex{p}_{z_0}\circ {\widehat{f}}_0{}^{-1}\circ Lo{g}_{z_0}\circ \dots \circ Ex{p}_{z_{k-1}}\circ {\widehat{f}}_{k-1}{}^{-1}\circ Lo{g}_{z_{k-1}}.\end{array}$$

Using an exponential map ${\phi }_i^{-1}=Ex{p}_{\mathrm{e.g}},$

φ _i = Log _{z i} , can be the inverse diffeomorphism ${\mathrm{\Phi }}_{\theta }^{-1}:y={\mathrm{e.g}}_k\mapsto {\mathrm{e.g}}_0=x$

be written as $$\begin{array}{l}{\mathrm{\Phi }}_{\theta }^{-1}=LO{G}_{{\mathrm{e.g}}_0}^{-1}\circ {\widehat{f}}_0{}^{-1}\circ Ex{p}_{{\mathrm{e.g}}_0}^1\circ \dots \circ LO{G}_{{\mathrm{e.g}}_{k-1}}^{-1}\circ {\widehat{f}}_{k-1}{}^{-1}\circ Ex{p}_{{\mathrm{e.g}}_{k-1}}^1\\ =Ex{p}_{{\mathrm{e.g}}_0}\circ {\widehat{f}}_0{}^{-1}\circ LO{G}_{{\mathrm{e.g}}_0}\circ \dots \circ Ex{p}_{{\mathrm{e.g}}_{k-1}}\circ {\widehat{f}}_{k-1}{}^{-1}\circ LO{G}_{{\mathrm{e.g}}_{k-1}}.\end{array}$$

dem Lösen der äquivalenten ODE vorwärts bzw. rückwärts in der Zeit im euklidischen Raum entsprechen, kollabiert das Differential dieser Abbildungen zu klassischen partiellen Ableitungen. Dies ermöglicht das wirksame Einsetzen des Algorithmus 1 im euklidischen Raum, um f̂_i und seine partiellen Ableitungen in einem einzigen Aufruf zu einem klassischen ODE-Löser zu erhalten. Zusammengefasst kann das Durchlaufen einer einzigen Komponente und die Berechnung ihres Differentials geschrieben werden als $$\begin{array}{c}{z}_i=\left(Ex{p}_{z_i}\circ {\widehat{f}}_i^{-1}\circ Lo{g}_{z_i}\right)z_{i+1}\\ D_{z_{i-1}}{z}_i=D_{y_i\left(t_s\right)}Ex{p}_{z_i}\circ D_{y_i\left(t_e\right)}{\widehat{f}}_i^{-1}\circ D_{z_{i-1}}Lo{g}_{z_i},\end{array}$$

wobei eine Exponential- und Logarithmusabbildung sowie ihre Ableitungen analytische Ausdrücke aufweisen. Das iterative Lösen der Gleichung (8) ergibt die Ergebnisse des Diffeomorphismus und des Pullback-Operators.There ${\widehat{f}}_i:y\left(t_s\right)\mapsto y_i\left(t_e\right)\text{and}{\widehat{f}}_i^{-1}:y_i\left(t_e\right)\mapsto y_i\left(t_s\right)$

correspond to solving the equivalent ODE forwards and backwards in time in Euclidean space, the differential of these maps collapses to classical partial derivatives. This allows leveraging Algorithm 1 in Euclidean space to obtain f̂ _i and its partial derivatives in a single call to a classical ODE solver. In summary, looping through a single component and computing its differential can be written as $$\begin{array}{c}{\mathrm{e.g}}_i=\left(Ex{p}_{{\mathrm{e.g}}_i}\circ {\widehat{f}}_i^{-1}\circ LO{G}_{{\mathrm{e.g}}_i}\right){\mathrm{e.g}}_{i+1}\\ D_{{\mathrm{e.g}}_{i-1}}{\mathrm{e.g}}_i=D_{y_i\left(t_s\right)}Ex{p}_{{\mathrm{e.g}}_i}\circ D_{y_i\left(t_e\right)}{\widehat{f}}_i^{-1}\circ D_{{\mathrm{e.g}}_{i-1}}LO{G}_{{\mathrm{e.g}}_i},\end{array}$$

where an exponential and logarithm map as well as their derivatives have analytical expressions. Iteratively solving equation (8) gives the results of the diffeomorphism and the pullback operator.

According to various embodiments, a method is summarized as in 5 illustrated provided.

5 shows a flowchart 500 illustrating a method for controlling a robotic device.

At 501, demonstrations (e.g., by a human user) are provided for a robot capability, each demonstration showing a trajectory that includes a sequence of robot configurations, each robot configuration being at least partially described by a point on a first Riemannian manifold.

At 502, a vector field is (pre)defined on a second Riemann manifold (homeomorphic to the first Riemann manifold, so that it can also be viewed as a second “instance” of the Riemann manifold).

At 503, a diffeomorphism between the first Riemannian manifold and the second Riemannian manifold is trained by searching for parameter values of the diffeomorphism that minimize a loss, including, for each demonstration and for at least one robot configuration of the trajectory shown by the demonstration, a difference between a configuration change to the next configuration of the trajectory and a configuration change given by the particular vector field via the diffeomorphism.

• • Abbilden des Punktes auf der ersten riemannschen Mannigfaltigkeit, der die Konfiguration beschreibt, auf einen Punkt auf der zweiten riemannschen Mannigfaltigkeit mittels des Diffeomorphismus;
• • Bestimmen einer Änderung, die durch das Vektorfeld an dem Punkt auf der zweiten riemannschen Mannigfaltigkeit spezifiziert wird; und
• • Abbilden der bestimmten Änderung auf den Tangentialraum des Punktes auf der ersten riemannschen Mannigfaltigkeit unter Verwendung des Differentials des inversen Diffeomorphismus. Das Ergebnis dieser Abbildung ist die Konfigurationsänderung, die durch das bestimmte Vektorfeld für eine Konfiguration über den Diffeomorphismus gegeben ist.

Here, determining the configuration change given by the particular vector field for a configuration involves

• • Mapping the point on the first Riemann manifold that describes the configuration to a point on the second Riemann manifold using diffeomorphism;
• • Determine a change specified by the vector field at the point on the second Riemann manifold; and
• • Mapping the particular change to the tangent space of the point on the first Riemann manifold using the differential of the inverse diffeomorphism. The result of this mapping is the configuration change given by the particular vector field for a configuration via diffeomorphism.

At 504, the robotic device is controlled for a configuration for which a control action is to be determined according to a configuration change given by the determined vector field via the trained diffeomorphism for the configuration for which a control action is to be determined (where the configuration change is according to can be determined from the above three bullet points).

Controlling the robot device according to a configuration change can be understood as controlling the robot device to perform the configuration change. For example, the configuration change specifies a speed in a certain direction, and the robotic device is controlled to move (or rotate) at the speed in that direction.

The Riemannian manifolds are, for example, spaces of points that specify an orientation of the robotic device, e.g. B. the orientation of an end effector.

The approach of 5 allows learning a control strategy (corresponding to a vector field on the first Riemannian manifold) via human demonstrations where path integrals (geodesics) are natural motion capabilities. The vector field is learned on the first Riemannian manifold in the sense that a diffeomorphism is learned that maps the desired (observed) demonstration data to the (pre)determined (stable) vector field on the second Riemannian manifold. Therefore, the global stability of the learned vector field can be guaranteed by the diffeomorphism applied to the (pre)determined vector field.

The procedure of 5 can be performed by one or more computers containing one or more data processing units. The term “data processing unit” can be understood as any type of entity that enables the processing of data or signals. For example, the data or signals may be treated according to at least one (ie, one or more than one) specific function performed by the data processing unit. A data processing unit may include an analog circuit, a digital circuit, a mixed signal circuit, a logic circuit, a microprocessor, a microcontroller, a central processing unit (CPU), a graphics processing unit (GPU), a digital signal processor (DSP), a programmable gate array (FPGA), include or be formed from an integrated circuit or any combination thereof. Any other way to implement the respective functions can also be understood as a data processing unit or logic circuit arrangement. It is understood that one or more of the method steps described in detail herein may be performed (e.g., implemented) by a data processing unit through one or more specific functions performed by the data processing unit.

Various embodiments may receive and use image data from various visual sensors (cameras), such as, for example, video, radar, LiDAR, ultrasound, thermal imaging, sonar, etc., to obtain the demonstration data.

The approach of 5 can be used to calculate a control signal for controlling a technical system, such as: B. a computer-controlled machine such as a robot, a vehicle, a household appliance, a power tool, a manufacturing machine, a personal assistant or an access control system. According to various embodiments, a strategy for controlling the technical system may be learned and then the technical system may be operated accordingly.

According to one embodiment, the method is computer-implemented.

Although specific embodiments have been illustrated and described herein, those of ordinary skill in the art will understand that a variety of alternative and/or equivalent implementations may be substituted for the specific embodiments shown and described without departing from the scope of the present invention. This application is intended to cover any adaptations or variations of the specific embodiments discussed herein. Therefore, it is intended that this invention be limited only by the claims and their equivalents.

Claims (8)

Method for controlling a robotic device, comprising: Providing demonstrations of a robotic capability, each demonstration showing a trajectory comprising a sequence of robot configurations, each robot configuration being at least partially described by a point on a first Riemannian manifold; determining a vector field on a second Riemannian manifold; Training a diffeomorphism between the first Riemannian manifold and the second Riemannian manifold by searching for parameter values of the diffeomorphism that minimize loss, including for each demonstration and for at least one robot configuration of the trajectory shown by the demonstration, a difference between a configuration change to the next configuration of the trajectory and a configuration change given by the particular vector field via the diffeomorphism, wherein determining the configuration change given by the determined vector field for a configuration includes mapping the point on the first Riemann manifold that describes the configuration to a point on the second Riemann manifold using diffeomorphism; determining a change specified by the vector field at the point on the second Riemann manifold; mapping the determined change to the tangent space of the point on the first Riemann manifold using the differential of the inverse diffeomorphism, where the configuration change is given by the result of the mapping; and Controlling the robotic device, for a configuration for which a control action is to be determined, according to a configuration change given by the determined vector field via the trained diffeomorphism for the configuration for which a control action is to be determined. Procedure according to Claim 1 , comprising determining that the vector field is a stable vector field. Procedure according to Claim 1 or 2 , comprising training the diffeomorphism by training a neural network, the diffeomorphism mapping a point to the solution of a neural ordinary differential equation, the value of the derivative at a point being given by the output of the neural network when fed the point . Procedure according to one of the Claims 1 until 3 , comprising determining the vector field on the second Riemannian manifold depending on the output of another neural network and training the further neural network together with the diffeomorphism by searching for parameter values of the diffeomorphism and the further neural network to minimize the loss. Procedure according to one of the Claims 1 until 4 , wherein each robot configuration includes an orientation of a component of the robot device described by the respective point on the first Riemann manifold. Robotic device controller designed to carry out the method according to one of the Claims 1 until 5 to carry out. Computer program comprising instructions which, when executed by a processor, cause the processor to carry out a method according to one of the Claims 1 until 5 carries out. A computer-readable medium that stores instructions that, when executed by a processor, cause the processor to perform a method according to one of the Claims 1 until 5 carries out.

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