Discrete-time Control Contraction Metrics (DCCM)

for Quasistatic Planar Pushing

using Smoothed Dynamics

MIT 2.152[J] Nonlinear Control

16 May 2023
 

By Shao Yuan Chew Chia

Harvard College

Control though Contact

It's important

Challenges:

  1. Non-smooth contact dynamics

  2. Underactuated system

Current Methods

Explicit enumeration of contact modes

Russ Tedrake. Underactuated Robotics: Algorithms for Walking, Running, Swimming, Flying, and Manipulation (Course Notes for MIT 6.832). Downloaded on May 14 from https://underactuated.csail.mit.edu/

Current Methods

Smoothing of contact dynamics

Our Approach

Smoothing of contact dynamics

+

Control Contraction Metrics

Control Contraction Metrics

Differential version of a Control Lyapunov Function

I. R. Manchester and J.-J. E. Slotine, “Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design,” IEEE Transactions on Automatic Control, vol. 62, no. 6, pp. 3046–3053, Jun. 2017, ISSN:1558-2523. DOI: 10.1109/TAC.2017.2668380.

Control Contraction Metrics

Advantages

  • Certificates of stability and convergence rates

  • Trajectory independent controllers

  • Convex synthesis of the controller

  • Faster online computation of the control law

Problem Setup

Dynamics:

  • Discrete-time
  • Analytically smoothed
  • Nonlinear
  • Control Affine
x = \begin{bmatrix}b_x & b_y & b_{\theta}& s_x& s_y\end{bmatrix}^\top
x_{k+1} = f(x_k) + g(x_k)u_k

Theory of DCCM

x_{k+1} = f(x_k) + g(x_k)u_k

dynamics

\delta_{x_{k+1}} = A(x_k)\delta_{x_k} + B(x_k)\delta_{u_k}

differential dynamics

A(x_k) = \frac{\partial (f(x_k) + g(x_k)u_k)}{\partial x_k} \in \mathbb{R}^{5 \times 5}
B(x_k) = \frac{\partial (f(x_k) + g(x_k)u_k)}{\partial u_k} \in \mathbb{R}^{5 \times 2}
\{

Jacobians

\delta_{u_k} = K(x_k)\delta_{x_k}

differential state feedback control law

Theory of DCCM

V_k = \delta^\top_{x_k} M_{k} \delta_{x_k}

differential squared distance in the positive definite metric \(M\)

\begin{aligned} V_{k+1} & = \delta^\top_{x_{k+1}} M_{k+1} \delta_{x_{k+1}} \\ & = \delta^\top_{x_k} (A_k + B_k K_k)^\top M_{k+1} (A_k + B_k K_k)\delta_{x_k} \end{aligned}

at the next time step...

V_{k+1} - V_k \leq - \beta V_k < 0 \quad \forall x, \delta_x \in \mathbb{R}^5, \beta \in (0,1]

contraction condition

Theory of DCCM (Online)

L(c) = \int_0^1 \sqrt{\frac{\partial{c(s)}}{\partial{s}} ^\top M(c(s)) \frac{\partial{c(s)}}{\partial{s}}} ds

Riemannian length

E(c) = \int_0^1 L(c)^2 ds

Riemannian energy

\begin{aligned} \gamma(x_0, x_1) &= \argmin_{c} L(c) = \arg \min_{c} E(c) \end{aligned}

geodesic

u_k = u^*_k + \int_0^1 K(\gamma(s))\frac{\partial{\gamma(s)}}{\partial{s}}

tracking controller

Methods: DCCM Synthesis

Sums of Squares (SOS) Programming

V_{k+1} - V_k \leq - \beta V_k < 0

contraction condition

\Rightarrow (A_k + B_k K_k)^\top M_{k+1} (A_k + B_k K_k) - (1 - \beta) M_k < 0

where \(W := M^{-1}\) and \(L := KW\)

\Rightarrow \begin{bmatrix} W_{k+1} & A_k + B_k L_k \\ (A_k + B_k L_k)^\top & (1 - \beta) W_k \end{bmatrix} > 0

Methods: DCCM Synthesis

Sums of Squares (SOS) Programming

\begin{aligned} \min_{l_c, w_c, r} \quad& r \\ s.t. \quad& \forall k, w^\top \Omega w - r w^\top w \in \Sigma(x_k, u_k, w) \\ & r \geq 0.1 \end{aligned}
\Omega = \begin{bmatrix} W_{k+1} & A_k + B_k L_k \\ (A_k + B_k L_k)^\top & (1 - \beta) W_k \end{bmatrix}

is SOS

relaxation slack variable

coefficients of \(L\) and \(W\)

enforced over samples

Methods: DCCM Synthesis

Sums of Squares (SOS) Programming

L_k = \begin{bmatrix} L_{11_k} & \cdots & L_{15_k} \\ L_{21_k} & \cdots & L_{25_k} \\ \end{bmatrix}
W_k = \begin{bmatrix} W_{11_k} & \cdots & W_{15_k} \\ \vdots & \ddots & \vdots \\ W_{15_k} & \cdots & W_{55_k} \\ \end{bmatrix}

matrix of polynomials

W.._{k} = w.._c v(x_k)
L.._k = l.._c v(x_k)

polynomial expression

v(x_k) = [x^4_{k_4}, x_{k_3}x^3_{k_4}, x^2_{k_3}x^2_{k_4}, \cdots ,x_{k_1}, x_{k_0}, 1]

monomial basis

Methods: DCCM Synthesis

Sampling strategy

Methods: Online Computation

Finding the geodesic

\forall i, \Delta s[i] > 0, \quad \sum_{i=0}^{N-1} \Delta s[i] = 1
\begin{aligned} \bar{\gamma}(x^*_k, x_k) = \argmin_{ \substack{ x[\cdot], \Delta x_{s}[\cdot],\\ \Delta s[\cdot]}} & \sum_{i=0}^{N-1} \Delta s[i] \Delta x_s[i]^\top M(x[i]) \Delta x_s[i] \\ \end{aligned}

geodesic energy

\begin{aligned} s.t. \quad x[0] = x^*_k, \quad x[N] = x_k \\ \end{aligned}

start and end of geodesic

\forall i, x[i+1] = x[i] + \Delta x_{s}[i]\Delta s[i]

continuity

Methods: Online Computations

Finding the geodesic

\forall i, \Delta s[i] > 0, \quad \sum_{i=0}^{N-1} \Delta s[i] = 1
\begin{aligned} \bar{\gamma}(x^*_k, x_k) = \argmin_{ \substack{ x[\cdot], \Delta x_{s}[\cdot],\\ \Delta s[\cdot], m[\cdot], y[\cdot]}} & \sum_{i=0}^{N-1} y[i] \end{aligned}
\begin{aligned} s.t. \quad x[0] = x^*_k, \quad x[N] = x_k \\ \end{aligned}
\forall i, x[i+1] = x[i] + \Delta x_{s}[i]\Delta s[i]

geodesic energy

\forall i, y[i] \geq \Delta s[i] \Delta x_s[i]^\top M(m[i]) \Delta x_s[i] \\
+ \Delta s[i]^2

even out length of segments

\( M = W^{-1}\)

\forall i, M(m[i]) W(x[i]) = I

Methods: Online Computations

Tracking controller

\begin{aligned} u_k = u^*_k + \sum_{i=0}^{N-1} \Delta s[i] K(x[i]) \Delta x_s[i] \end{aligned}
= u^*_k + \sum_{i=0}^{N-1} \Delta s[i] L(x[i]) W^{-1}(x[i]) \Delta x_s

Results

Overview

Results

Controller performance

Results

Effect of number of samples on performance

Results

Unexplained issues

 

  • Discretization of the geodesic
    • Optimization program failed when \(N > 1\)
  • Choosing \(\beta\)
    • Synthesizing DCCM with \(\beta > 0.3\) failed
    • DCCM with \(\beta = 0.3\) did not stabilize the system while \(\beta = 0.1\) did

Results

Computation time

 

Task Parameters Time
Synthesis Deg 4, 500 Samples 18 min
Synthesis Deg 6, 500 Samples 3 hr 20 min
Synthesis Deg 4, 2000 Samples 1 hr
Geodesic Deg 4, 1 Segment 1.54 s
Geodesic Deg 6, 1 Segment 3.81 s

Linux machine with 31.3GB of RAM, an with an Intel Core i7-6700 CPU @ 3.40GHz x 8 processor.

Conclusion and Future Work

Success:

  • Synthesized stabilizing DCCM with highly smoothed dynamics for planar pushing

Yet to be achieved:

  • Certificates of stability and convergence rates
    • Error due to sparse sampling
    • Error due to exact vs smoothed dynamics
  • Trajectory independent controller
  • Synthesis of controller for contact dynamics with less smoothing
    • Learn the metric [13], [14]
  • Faster computation of the control law
    • Warm starting techniques
    • Pseudospectral approaches [15]

Other possible directions: stabilizing to submanifold [8]

References

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2023). Global Planning for Contact-Rich Manipulation
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to this work. arXiv: 2206 . 10787 [cs], [Online].
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[2] D. E. Stewart and M. Anitescu, “Optimal control of
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[3] F. R. Hogan and A. Rodriguez. (Nov. 24, 2016). Feed-
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[4] J.-P. Sleiman, J. Carius, R. Grandia, M. Wer-
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[5] K. Nakatsuru, W. Wan, and K. Harada. (Feb. 25, 2023).
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[8] I. R. Manchester and J.-J. E. Slotine, “Control Contrac-
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[15] K. Leung and I. R. Manchester. (Nov. 4, 2017). Non-
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