Moving fast to contact

Introduction

Consider the following problem: a position control robot, moving to make contact with a stiff, spring-like environment with stiffness \(K\),

Deriving the optimal contact force

Consider the following naive, but optimal, strategy: Robot moves down with velocity \(v\), and stops immediately when it measures a small force. At this instance, the robot has actually been in contact with the surface for \(T\) seconds. Following this strategy, the robot will stop exactly \(T\) seconds after it actually is in contact, the contact force is therefore \[ F_{\max} = T v K \]

Suppose \(T\) is 10ms, \(K\) is 50 N/mm we have \[ F_{\max} = 0.01 * 50 v = 0.5v \] where \(v\) is in mm per seconds. That is to say if the robot moves down at 20 mm/s, the smallest force that it can receive is 10N.

Summary: If the time-delay is 10ms, the smallest possible force one can obtain for every 1mm/s of velocity is 0.5N

Is this strategy reasonable? Notice that the robot has to make the decision to stop even when the measured force is very small. If the robot is actually under a linear control law, which usually produce small input command given small force measurement, this means the gain must be very large. In most case, this would lead to instability. This argument applies too all linear controllers.

Relevant to 2002SwitchingBetweenStabilizingControllers.

Linear control law

It’s interesting to see what is the maximum force given a velocity command. We can try out some simulation. Suppose the robot is under a linear control law: \[ v = K_g (f_d - f) + v_d.\]

The system (robot/environment) dynamics is given by two equations: \[ x = e^{-T s} \frac{v}{s}, f = K x \] We can then simulate the instance to find the maximum force exerted on the robot as it moves down.

import control as co
import matplotlib.pyplot as plt
import numpy as np

s = co.tf([1, 0], [1])
z = co.tf([1, 0], [1], 1e-3)
# x = G v
G = z ** (-10) * co.c2d(1 / s, 1e-3)


ymax = []
gains = np.arange(0.1, 3.0, 0.1)
for gain in gains:
    # f * H = G g f_d + v_d
    H = 1 / 50.0 + G * gain
    # f = M v_d
    M = G / H
    t, y = co.step_response(M, T=np.arange(0, 1, 0.001))

    # plt.plot(y)
    # plt.show()
    ymax.append(np.max(y))
plt.plot(gains, ymax)
plt.xlabel('gain')
plt.ylabel("max force amplitude")
plt.grid(True)
plt.savefig('gain-force-compare.png')
plt.show()

Figure 1: Maximum reaction force when moving at 1mm/s at different linear gain

Figure 1: Maximum reaction force when moving at 1mm/s at different linear gain

It should be noted that at gain around 2.0, the system exhibit dynamic instability.

From this graph, it’s clear that the more reponsive the robot is, the smaller force it feels. This is quite intuitive. What is perharp less intuitive is that if the gain is too small, the robot might feels a very large force because it’s not fast enough to react to the measurement.

CCS?

Optimizing a controller specifically for reducing the maximum force using ConvexControllerSynthesis is an interesting idea. I tried it out and manage to reach approximately 1N minimum force. The resulting controller, however, is not at all optimized for other objectives. Thus, it is not suitable for other purposes.

A high-level control law

From the previous two sections, it seems like designing a single controller that allows fast movement in freespace and stability and performance in contact is simply impossible.

Applying a high-level control law, on the other hand, can achieve high-performance contact very easily. Consider the following simple high-level control law:

  1. move down with speed 20mm/s, f_d = 0;

  2. when in contact immediately switch velocity back to 0mm/s and f_d = 1N;

  3. similarly, we can adjust the reference velocity based on the measure force. For example we can have:

    if f > 5: v_d = 0
    else: v_d = 4 * (5 - f)
    

This control law gonna work just fine without much trouble. Putting together with a linear controller, we thus have a simple non-linear control law that will perform relatively well.

Conclusion

Even for the very simple task of making contact as fast as possible, we learn that a high-level control strategy is indispensible. Low-level control laws, i.e. CCS or linear controllers, are insufficient to guarantee high performance.

For more complex task, such as 1 pin insertion, 2 pins insertion slanted pin, it is clear that a good high-level control strategy is even more important. Thus it would be very interesting to study this question more closely.

Question for future inquiry:

ConvexControllerSynthesis

In this note I solve this problem using RL RlTrialMoveFastToContact.