Human-Machine Skill Transfer Method Based on Geometric Perception and Rhythmic Dynamic Motion Primitives

Abstract

The present invention discloses a human-robot skill transfer method based on geometric perception and rhythmic dynamic motion primitives, including: S1. constructing a desired behavior model of the robot; S2. estimating a stiffness matrix based on the desired behavior model of the robot to obtain an estimated stiffness matrix; S3. constructing a robot skill learning model based on geometric perception and rhythmic dynamic motion primitives; S4. inputting the estimated stiffness matrix into the robot skill learning model, so that the robot imitates the stiffness trajectory and realizes the reproduction of the stiffness trajectory. The present invention enables the robot to more simply, stably, accurately and smoothly learn motion skills with a symmetric positive definite (SPD) matrix manifold structure such as impedance.

Description

Technical Field

[0001] The present invention relates to the field of collaborative robots, and in particular to a human-machine skill transfer method based on geometric perception and rhythmic dynamic motion primitives. Background Art

[0002] The ability to reliably perform assigned tasks in highly unstructured and dynamic environments is fundamental to bringing robots into our daily lives. To do this, the robot needs to accurately control its movements in free space and during physical interactions, which requires it to be able to generate and adapt motion, impedance and / or force. Therefore, human expertise can be leveraged to teach robots how to perform such tasks by transferring human skills to them. Learning from human demonstrations (LfD) has been widely studied as a convenient way to transfer human skills to robots. The learning method aims to extract relevant movement patterns from human demonstrations and subsequently apply these patterns to different situations.

[0003] Dynamic motion primitives (DMPs) are a widely used LfD method with many beneficial properties, such as robustness against disturbances and the ability to adapt to new requirements. Many tasks in these dynamic environments require variable impedance, especially in some rhythmic manipulation tasks such as blackboard erasing, sawing, and water pumping. However, the impedance parameters are encapsulated in symmetric positive definite (SPD) matrices. Since traditional rhythmic DMPs (Dynamic Movement Primitives) rely on the Euclidean parameterization of space, the variable impedance skill parameters for these SPD matrix manifold structures cannot be used directly, resulting in the robot being unable to stably and smoothly learn motion skills with symmetric positive definite (SPD) matrix manifold structures. Therefore, a human-machine skill transfer method based on geometric perception and rhythmic dynamic motion primitives is needed to solve the above problems. Summary of the invention

[0004] In view of this, the purpose of the present invention is to overcome the defects in the prior art and provide a human-machine skill transfer method based on geometric perception and rhythmic dynamic motion primitives, which can enable the robot to learn motion skills with a symmetric positive definite (SPD) matrix manifold structure such as impedance more simply, stably, accurately and smoothly.

[0005] The human-machine skill transfer method based on geometric perception and rhythmic dynamic motion primitives of the present invention comprises the following steps:

[0006] S1. Construct the robot's expected behavior model;

[0007] S2. Estimating the stiffness matrix based on the expected behavior model of the robot to obtain an estimated stiffness matrix;

[0008] S3. Construct a robot skill learning model based on geometric perception and rhythmic dynamic motion primitives;

[0009] S4. Input the estimated stiffness matrix into the robot skill learning model, so that the robot imitates the stiffness trajectory and achieves the reproduction of the stiffness trajectory.

[0010] Furthermore, the robot's expected behavior model is determined according to the following formula:

[0011]

[0012] Among them, M is the mass matrix, B is the damping matrix, K is the stiffness matrix, X d is the desired position of the robot, X is the current position of the robot, is the current speed of the robot, is the current acceleration of the robot, F e The interaction force between the robot and the outside world.

[0013] Further, the step S2 specifically includes:

[0014] S21. Linearize the robot's expected behavior model to obtain a linearized model:

[0015]

[0016] in,

[0017] S22. Solve the linearized model using the least squares method to obtain an estimate of the stiffness matrix at each moment;

[0018] S23. An optimization model is introduced to optimize the solved linearized model to obtain a stiffness matrix that satisfies the symmetric positive definite matrix constraint; the optimization model is:

[0019]

[0020] in, is the estimated stiffness matrix, H is the singular value decomposition of P P = USV T The symmetric polarity factor of , U and V are unitary matrices, and S is a diagonal matrix containing singular values.

[0021] Furthermore, the robot skill learning model is determined according to the following formula:

[0022]

[0023] Among them, τ is the reciprocal of the operating frequency Ω, is the second-order derivative of the stiffness matrix K, α y and βy are system gain parameters; K g is the target symmetric positive definite matrix, K j is time t j The corresponding stiffness matrix, K 1 is the initial stiffness matrix,

[0024] z is the first-order derivative of the stiffness matrix K, vec(·) is a function that transforms a symmetric matrix into a vector using the Mandel symbol;

[0025]

[0026] Ψ i (φ)=exp(h(cos(Ψ i (φ)-c i )-1));

[0027] Where N is the total number of data points and φ is the phase; is the weight, r is the modulation period signal, c i and h are the basis functions Ψ i The center and width of (φ).

[0028] Furthermore, the phase and frequency τ are estimated according to the following formula:

[0029]

[0030]

[0031] in, is the first-order derivative of the phase φ, is the first-order derivative of frequency Ω, Ω = 1 / τ, P is a positive definite coupling constant, is the external signal U and the internal estimate The difference between

[0032]

[0033] Where M is the number of Fourier series, c is the sequence number of the Fourier series,

[0034] A and B are constants, η is the learning rate;

[0035]

[0036] Furthermore, the robot is made to imitate the stiffness trajectory according to the following formula:

[0037]

[0038] in, is the stiffness matrix simulated by the robot, t represents the current moment, δt is the time interval, Exp K(t) (·) represents the exponential mapping function, K(t) is the stiffness matrix at the current moment, z(t) is the vector representation of the stiffness matrix at the current moment, represents the parallel transfer function, K 1 is the initial stiffness matrix, mat(·) is the function that transforms the vector into a symmetric matrix using the Mandel symbol, and τ is the inverse of the operating frequency Ω.

[0039] The beneficial effects of the present invention are as follows: a human-machine skill transfer method based on geometric perception and rhythmic dynamic motion primitives disclosed by the present invention first uses the least squares algorithm to identify the teaching impedance model through the force-position characteristic parameters collected during human kinesthetic teaching, and standardizes it through SPD matrix processing; secondly, considering the particularity of the impedance skill symmetric positive definite (SPD) matrix manifold structure, it is integrated into the rhythmic DMPs control framework based on Riemann manifold geodesics, logarithmic / exponential mapping, and parallel transmission methods, so that the robot can learn variable impedance skills simply, stably, accurately and smoothly, and realize efficient transfer of human-machine skills. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] The present invention will be further described below in conjunction with the accompanying drawings and embodiments:

[0041] Figure 1 It is a schematic diagram of the principle of the skill transfer method of the present invention;

[0042] Figure 2 A schematic diagram of the difference between the geodesic and Euclidean paths between two SPD matrices of the present invention;

[0043] Figure 3 The SPD manifold of the present invention is Schematic diagram of exponential, logarithmic mapping and parallel transmission on;

[0044] Figure 4 A schematic diagram of the distance between the stiffness profile obtained by the skill transfer method of the present invention and the true value;

[0045] Figure 5 The first and second derivatives generated for the proposed geometry-aware rhythm DMPs are used to learn the stiffness ellipsoid schematic;

[0046] Figure 6 Schematic diagram of the SPD stiffness matrix in the embedded conic section space for the teaching and reproduction of the present invention. DETAILED DESCRIPTION

[0047] The present invention is further described below in conjunction with the accompanying drawings, as shown in the drawings:

[0048] The human-machine skill transfer method based on geometric perception and rhythmic dynamic motion primitives of the present invention comprises the following steps:

[0049] S1. Construct the robot's expected behavior model;

[0050] S2. Estimating the stiffness matrix based on the expected behavior model of the robot to obtain an estimated stiffness matrix;

[0051] S3. Construct a robot skill learning model based on geometric perception and rhythmic dynamic motion primitives;

[0052] S4. Input the estimated stiffness matrix into the robot skill learning model, so that the robot imitates the stiffness trajectory and achieves the reproduction of the stiffness trajectory.

[0053] The present invention aims at the particularity of symmetric positive definite (SPD) matrix manifold structures such as impedance and the necessity of rhythmic motor skills in daily life (such as blackboard erasing, sawing and pumping water, etc.) for characteristic parameter learning. At the same time, it pays attention to the dependence of traditional rhythmic dynamic motion primitives on Euclidean space parameters, and forms a human-machine skill transfer analysis method combining geometric perception and rhythmic dynamic motion primitives, so that robots can learn motor skills such as impedance with symmetric positive definite (SPD) matrix manifold structures more simply, stably, accurately and smoothly, which has technical guiding significance for the research on robot skill learning.

[0054] In this embodiment, in step S1, when the operator performs kinesthetic teaching, the desired behavior of the robot end effector is modeled through robot impedance control to obtain the robot desired behavior model:

[0055]

[0056] Among them, M, B and K are mass matrix, damping matrix and stiffness matrix respectively, which are dynamic parameters; X d is the desired position of the robot, X is the current acceleration, velocity and position of the robot, F e The interaction force between the robot and the outside world.

[0057] In this embodiment, in step S2, a sliding window technique is used to achieve local stiffness estimation that satisfies the above formula (1) at each time step, and a window with a length of L is drawn along the demonstration data. X and F e Move. Define all data in the window as M and B are both predefined values, and the following linearized model can be obtained:

[0058]

[0059] The least squares method is used to solve the above linearized model and estimate the stiffness matrix at each moment. is a symmetric positive definite (SPD) matrix, which means that the solution obtained by the least squares method is only a rough estimate that does not satisfy the SPD constraint.

[0060] Therefore, the following optimization model is introduced to convert the above calculated approximate results into the nearest SPD matrix. The optimization model is:

[0061]

[0062] in, is the estimated stiffness matrix, H is the singular value decomposition of P P = USV T The symmetric polarity factor of , U and V are unitary matrices, and S is a diagonal matrix containing singular values.

[0063] At the same time, since some approximation matrices may be on the boundary of the SPD matrix space, a symmetric semi-positive definite matrix is ​​obtained. In this case, the stiffness matrix estimate Each eigenvalue λ of i ≈0 is constrained by the minimum lower bound, and then its eigenvalue decomposition is used To reconstruct The damping term of the impedance equation can be obtained experimentally or using The eigendecomposition of is chosen to keep the critical damping of the system, such as Among them, Q and Δ are The eigenvectors and eigenvalues ​​of is the damping ratio.

[0064] In this embodiment, in step S3, due to the stiffness matrix It is not closed under addition and scalar product, so it cannot be considered as a vector space, and therefore it is not sufficient to use classical Euclidean space methods to process and analyze these matrices.

[0065] Construct a robot skill learning model based on geometric perception and rhythmic dynamic motion primitives, including:

[0066] The SPD matrix is ​​combined with the Riemannian metric to form a Riemannian manifold, and the geodesic metric is defined, that is, the minimum length curve between two points on the manifold, such as Figure 2 shown.

[0067] Riemannian manifold is a topological space, each point of which is locally similar to Euclidean space. There exists a tangent space The metric in the tangent space is flat, allowing the use of classical arithmetic tools such as Figure 3 shown.

[0068] For an SPD manifold, any point The tangent space at is given by the symmetric matrix space Sym D (D represents the dimension). The space of SPD matrices can be represented as embedded in the tangent space Sym D The interior of a convex cone.

[0069] Exponential Mapping Exp Σ : It means mapping the point L on the tangent space to the point Λ on the manifold.

[0070] Logarithmic Mapping Σ : It means mapping the point Λ on the manifold to the point L on the tangent space. And there is the following formula:

[0071]

[0072]

[0073] Parallel transmission Γ Σ→Λ : It is used to move elements in the tangent space so that the angle between them remains unchanged. And there is the following formula:

[0074]

[0075] in, arrive

[0076] definition As a demonstration set of SPD matrix, j represents the time and T represents the total time.

[0077] Secondly, the first-order and second-order derivatives of the stiffness matrix K are solved and substituted into the rhythmic dynamic motion primitive model.

[0078] The first-order derivative is as follows:

[0079]

[0080] In order to avoid information duplication caused by symmetry and occupying computing resources, the Mandel representation method is used to reduce the data dimension to m=D+D(D-1) / 2.

[0081] Introduce the following formula:

[0082]

[0083] where vec(·) is a function that transforms a symmetric matrix into a vector using the Mandel symbol;

[0084]

[0085] The second-order derivative is directly computed using standard Euclidean tools and is vectorized as Then we get the data set: The data set includes: at each time t j , t j The corresponding stiffness matrix K j , and the stiffness matrix K j The first and second derivatives of z j and

[0086] Therefore, the robot skill learning model combining geometric perception and rhythmic dynamic motion primitives is expressed as follows:

[0087]

[0088] Among them, τ is the reciprocal of the operating frequency Ω, is the second-order derivative of the stiffness matrix K, α y and β y All are system gain parameters; is the target symmetric positive definite matrix, which is obtained by the average value of the data set (all matrices K); K j is time t j The corresponding stiffness matrix, K 1 is the initial stiffness matrix,

[0089]

[0090] z is the first-order derivative of the stiffness matrix K, vec(·) is a function that transforms a symmetric matrix into a vector using the Mandel symbol;

[0091] is the forcing term, expressed as follows:

[0092]

[0093] Ψ i (φ)=exp(h(cos(Ψ i (φ)-c i )-1))

[0094] Where N is the total number of data points, that is, the total number of data in the data set, φ is the phase, and r is used to modulate the periodic signal (when r = 1, the rhythmic motion is not scaled). iand h is the basis function Ψ i The center and width of (φ). Among them, c i Uniformly distributed in [0,2π], h is set to 2.5 times the number of basis functions. It can be obtained by using local weighted regression learning.

[0095] The phase φ and frequency Ω are estimated by the adaptive oscillator, which can be expressed as follows:

[0096]

[0097]

[0098] in, is the first-order derivative of the phase φ, is the first-order derivative of frequency Ω, Ω = 1 / τ, P is a positive definite coupling constant, is the external signal U and its internal estimate The difference between them is constructed by Fourier series. In this embodiment, U is the input stiffness matrix K.

[0099]

[0100] Where M is the number of Fourier series, c is the sequence number of the Fourier series,

[0101] A and B are both constants, specifically:

[0102]

[0103] The learning method of Fourier series parameters is as follows:

[0104]

[0105]

[0106] where η is the learning rate. Adaptive oscillators are most useful for inferring periodic states (phase and frequency) in real time. However, they are also useful for offline learning when the recorded signal frequency is variable. Figure 6 As shown, the dotted line represents the representation of the reproduced SPD stiffness matrix in the embedded conic tangent space, and the solid line represents the representation of the taught SPD stiffness matrix in the embedded conic tangent space.

[0107] In this embodiment, in step S4, the constructed robot skill learning model is used to enable the robot to learn the estimated stiffness matrix and achieve the reproduction of the stiffness trajectory.

[0108] In the reproduction stage, the robot imitates the stiffness trajectory according to the following formula to obtain the simulated stiffness trajectory:

[0109]

[0110] Among them, mat(·) is the inverse of vec(·), that is, a function that transforms a vector into a symmetric matrix using the Mandel symbol, indicating matrixization using the Mandel symbol. is the stiffness matrix simulated by the robot, t represents the current moment, δt is the time interval, Exp K(t) (·) represents the exponential mapping function, K(t) is the stiffness matrix at the current moment, z(t) is the vector representation of the stiffness matrix at the current moment, represents the parallel transfer function, K 1 is the initial stiffness matrix, Represents a new SPD matrix based robot impedance technology.

[0111] Aiming at the necessity of learning variable impedance skills in daily life, the present invention first designs an impedance model identification strategy with a symmetric positive definite (SPD) matrix manifold structure through a least squares algorithm; then combines Riemann manifolds with rhythmic dynamic motion primitives to form a geometrically perceived human-machine skill transfer method, which fills the gap in the current robot's variable impedance skill learning with a symmetric positive definite (SPD) matrix manifold structure. At the same time, based on the dynamic motion primitive algorithm, it can be integrated with traditional position and posture dynamic motion primitives to form a unified skill expression, and can also be extended to skill learning aspects such as operability, which has important technical guiding significance in actual work and future development.

[0112] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solution of the present invention, which should be included in the scope of the claims of the present invention.

Claims

1. A human-machine skill transfer method based on geometric perception and rhythmic dynamic motion primitives, Features: The steps include: S1. Construct the robot's expected behavior model; determine the robot's expected behavior model according to the following formula: Among them, M is the mass matrix, B is the damping matrix, K is the stiffness matrix, X d is the desired position of the robot, X is the current position of the robot, is the current speed of the robot, is the current acceleration of the robot, F e is the interaction force between the robot and the outside world; S2. Estimating the stiffness matrix based on the expected behavior model of the robot to obtain the estimated stiffness matrix; the step S2 specifically includes: S21. Linearize the robot's expected behavior model to obtain a linearized model: in, S22. Solve the linearized model using the least squares method to obtain an estimate of the stiffness matrix at each moment; S23. An optimization model is introduced to optimize the solved linearized model to obtain a stiffness matrix that satisfies the symmetric positive definite matrix constraint; the optimization model is: in, is the estimated stiffness matrix, H is the singular value decomposition of P P = USV T The symmetric polarity factor of , U and V are unitary matrices, and S is a diagonal matrix containing singular values; S3. Construct a robot skill learning model based on geometric perception and rhythmic dynamic motion primitives; The robot skill learning model is determined according to the following formula: Among them, τ is the reciprocal of the operating frequency Ω, is the second-order derivative of the stiffness matrix K, α y and β y are system gain parameters; K g is the target symmetric positive definite matrix, K j is time t j The corresponding stiffness matrix, K 1 is the initial stiffness matrix, z is the first-order derivative of the stiffness matrix K, vec(·) is a function that transforms a symmetric matrix into a vector using the Mandel symbol; P i (φ)=exp(h(cos(Ψ) i (φ)-c i )-1)); Where N is the total number of data points and φ is the phase; is the weight, r is the modulation period signal, c i and h are the basis functions Ψ i The center and width of (φ); S4. Inputting the estimated stiffness matrix into the robot skill learning model, so that the robot imitates the stiffness trajectory and realizes the reproduction of the stiffness trajectory; The robot simulates the stiffness trajectory according to the following formula: in, is the stiffness matrix simulated by the robot, t represents the current moment, δt is the time interval, Exp K(t) (·) represents the exponential mapping function, K(t) is the stiffness matrix at the current moment, z(t) is the vector representation of the stiffness matrix at the current moment, represents the parallel transfer function, K 1 is the initial stiffness matrix, mat(·) is the function that transforms the vector into a symmetric matrix using the Mandel symbol, and τ is the inverse of the operating frequency Ω.

2. The human-machine skill transfer method based on geometric perception and rhythmic dynamic motion primitives according to claim 1, Features: The phase and frequency τ are estimated according to the following formula: in, is the first-order derivative of the phase φ, is the first-order derivative of frequency Ω, Ω = 1 / τ, P is a positive definite coupling constant, is the external signal U and the internal estimate The difference between Where M is the number of Fourier series, c is the sequence number of the Fourier series, A and B are constants, η is the learning rate;

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