Subsections

• Formulation
• Implementation
• Performance

8 Robot Arm

Minimize the time taken for a robot arm to travel between two points.

Formulation

This problem originated in the thesis of Monika Mössner-Beigel (Heidelberg University). In her formulation the arm of the robot is a rigid bar of length that protrudes a distance from the origin to the gripping end and sticks out a distance in the opposite direction. If the pivot point of the arm is the origin of a spherical coordinate system, then the problem can be phrased in terms of the length $\rho$ of the arm from the pivot, the horizontal and vertical angles $( \theta , \phi )$ from the horizontal plane, the controls $( u_{\rho}, u_{\theta}, u _{\phi} )$, and the final time $t_f$. Bounds on the length and angles are

$$\rho (t) \in [0, L ] , \qquad \vert \theta (t) \vert \le \pi , \qquad 0 \le \phi (t) \le \pi ,$$

and for the controls,

$$\vert u_{\rho} \vert \le 1 , \qquad \vert u_{\theta} \vert \le 1, \qquad \vert u_{\phi} \vert \le 1 .$$

The equations of motion for the robot arm are

$$L {\rho'' } = u_{\rho } , \qquad I_{\theta } {\theta''} = u_{\theta }, \qquad I_{\phi } {\phi'' } = u_{\phi } ,$$ (5)

where is the moment of inertia, defined by

$$I_{\theta } = \frac{((L-\rho )^{3}+\rho ^{3})}{3}\sin (\phi )^{2}, \qquad I_{\phi } = \frac{((L-\rho )^{3}+\rho ^{3})}{3} .$$

The boundary conditions are

$$\rho (0) = \rho (t_{f}) = 4.5 , \qquad \theta (0) = 0 , \ \t... ...2\pi }{3} , \qquad \phi (0) = \phi (t_{f}) = \frac{\pi }{4} ,$$

$${\rho'}(0) = {\theta'}(0) = {\phi'}(0) = {\rho'}(t_{f}) = {\theta'}(t_{f}) = {\phi'}(t_{f}) = 0.$$

This model ignores the fact that the spherical coordinate reference frame is a noninertial frame and should have terms for Coriolis and centrifugal forces.

Implementation

In the implementation of Vanderbei [22] the controls $u$ are eliminated by substitution, and thus the equality constraints in (8.1) become the inequalities

$$\vert L {\rho '' } \vert \le 1, \qquad \vert I_{\theta } {\t... ...'} \vert \le 1, \qquad \vert I_{\phi } {\phi ''} \vert \le 1 .$$

In this implementation is expressed in terms of a first-order system with the additional variables $\rho '$, $\theta '$, and $\phi '$. Discretization is done with a uniform time step and the trapezoidal rule over intervals. Data for this problem is shown in Table 8.1.

Table 8.1: Robot arm problem data

Variables
Constraints
Bounds
Linear equality constraints	0
Linear inequality constraints	0
Nonlinear equality constraints
Nonlinear inequality constraints	0
Nonzeros in	0
Nonzeros in

Performance

Results for the AMPL implementation appear in Table 8.2. All solvers were given the same initial values. The initial values for $\rho$ and $\phi$ were set to the functions $\rho \equiv 4.5$ and $\phi \equiv \pi /4$ evaluated at the grid points. Similarly, initial values for $\theta$ were set to the discrete version of the parabola

$$\theta (t ) = \frac{2 \pi } {3} \left ( \frac {t}{t_f} \right ) ^ 2 ,$$

which matches three of the boundary conditions. The initial values for all the controls were set to zero, and $t_f = 1$ initially.

Table 8.2: Performance on robotic arm problem

Solver	$n_h=50$	$n_h=100$	$n_h=200$	$n_h=400$
LANCELOT	0.47 s	0.53 s	1.2 s	2.59 s
$f$	0.00000e+00$\dagger$	-2.77555e-17$\dagger$	-2.77555e-17$\dagger$	-2.77555e-17$\dagger$
$c$ violation	4.18880e-02$\dagger$	2.09440e-02$\dagger$	1.04720e-02$\dagger$	5.23600e-03$\dagger$
iterations	5	3	3	3
LOQO	1.03 s	2.77 s	$\ddagger$	$\ddagger$
$f$	9.14687e+00	9.14267e+00	$\ddagger$	$\ddagger$
$c$ violation	2.4e-10	4.7e-11	$\ddagger$	$\ddagger$
iterations	24	30	$\ddagger$	$\ddagger$
MINOS	2.82 s	9.89 s	37.96 s	161.18 s
$f$	9.14687e+00	9.14267e+00	9.14138e+00	9.14108e+00
$c$ violation	2.0e-13	1.0e-10	2.7e-12	5.7e-13
iterations	9	13	22	37
SNOPT	10.22 s	30.7 s	317.52 s	2671.63 s
$f$	9.14687e+00	1.92751e+01$\dagger$	9.14142e+00	9.14101e+00
$c$ violation	1.9e-10	3.4e-04$\dagger$	1.4e-10	2.1e-10
iterations	24	9	31	29
$\dagger$ Errors or warnings. $\ddagger$ Timed out.

LANCELOT reports that it could not find a feasible solution for any of the versions we try for this implementation. For $n_h = 100$, SNOPT encounters difficulties, which it describes as an error evaluating nonlinear expressions.

Figure 8.1 shows the variables $\rho$, $\theta$, $\phi$ for the robot arm as a function of time. We also show in Figure 8.2 the controls $u_{\rho}$, $u_{\theta}$, $u _{\phi}$ as a function of time. Note that the controls for the robot arm are bang-bang. Also note that the functions $\rho$, $\theta$, $\phi$ for the robot arm are continuously differentiable, but since the second derivatives are directly proportional to the controls, the second derivatives are piecewise continuous.

Figure 8.1: Variables $\rho$, $\theta$, $\phi$ for the robot arm as a function of time

Figure 8.2: Control variables $u_{\rho}$, $u_{\theta}$, $u _{\phi}$ for the robot arm as a function of time