Knauer, Matthias; Buskens, Christof

Real-Time Trajectory Planning of the Industrial Robot IRB6400

Path planning of industrial robots requires the calculation of trajectories with respect to different optimization criteria,e.g. time optimal, energy optimal or wear and tear optimal solutions. These optimal control problems can be solvednumerically, e.g., by direct solvers. In practice, inaccuracies or perturbations in the underlying model can cause therobot to fail in following the calculated path. When deviations of the precalculated trajectory or system parametersare detected an update of the robot trajectory is required in real-time. Therefore, a method based on the parametricsensitivity analysis of the optimal control problem is proposed to calculate admissible approximations for the solutionof the perturbed robot trajectory problem in real-time. A model of ABB’s common robot IRB 6400 is used to revealthe capabilities of the proposed method.

1. Introduction

Industrial robots are widely used in industry. The IRB6400, a robot manufactured by ABB Flexible Automationwith more than 20.000 units so far, is typically used for spot welding tasks in the automobile industry. It featuresa low energy consumption due to a balanced construction consisting of a counterweight of about half a ton and twopneumatic cylinders, so that it can move payloads up to 200 kg with 5m/s as maximum speed.

2. Optimal Trajectory Problem

The efficiency of industrial facilities rises with the independence of trajectory planning. However, selecting anappropriate objective function F (q, u) still requires some experience and depends on the particular task. Theoptimal trajectory problem for the industrial robot can be stated as follows:

minu

F (q, u) = time, energy, . . .

s. t. M(q)q = u + v(q, q) + w(q), u ∈ [umin, umax]q(0) = q0, q(tf ) = qf , q ∈ [qmin, qmax]q(0) = q0, q(tf ) = qf , q ∈ [qmin, qmax]

(1)

Modelling the robot as a multi body rigid system results in an ODE system for the set of joint angles q. The motorcapacity of the joints is the control variable u. M denotes the symmetric and positive definite mass matrix, whichrepresents the relative translations and rotations of the rigid bodies in the system. v contains the centrifugal andCoriolis forces, w the gravitational forces. Furthermore the optimal solution has to fulfil boundary conditions aswell as state and control constraints. In this paper we will discuss a system with only three degrees of freedomcorresponding to the first three rotatoric joints regarding the positioning of the end effector, cf. Figure 1.

3. Receiving Trajectory Data for Real-Time Optimization

An optimal solution of the trajectory problem as given above can be found by using the direct solver NUDOCCCS[1] which approximates the infinitely dimensional optimal control problem by a non-linear optimization problem.

minz∈IRn

F (z, p)

s. t. G(z, p) = 0(2)

In order to apply real-time strategies, a parametric sensitivity analysis for the optimal solution provides informationon how the solution reacts on deviations p ∈ IRnp in the system data.

The sensitivity information for the unperturbed problem with fixed p = p0 can be calculated as(dzdp (p0)dλdp (p0)

)= −

(Lzz GT

z

Gz 0

)−1(Lzp

Gp

), (3)

PAMM · Proc. Appl. Math. Mech. 3, 515–516 (2003) / DOI 10.1002/pamm.200310527

e.g., if strong second order sufficient conditions hold. Herein L(z, λ, p) := F (z, p) + λT G(z, p) denotes the Lagrangefunction with Lagrange multiplier λ ∈ IRm.

4. Real-Time Optimal Control Using Sensitivity Information

As changes in the robot’s dynamics, e.g. deviations in the initial values, lead to inaccuracies in the solution and asthe process times for robot movements are only a few seconds, we need an online optimization of the trajectory. Ifa deviation ∆p from p0 is detected (p = p0 + ∆p) the first order Taylor expansion

z(p) ≈ z[1] := z(p0) +dz

dp(p0)(p − p0) (4)

yields a first approximation for the disturbed optimal solution z(p). Checking the constraints, we neither canexpect that G(z[1](p), p) = 0 holds nor that z[1](p) is optimal. Anyhow we find ‖G(z[1](p), p)‖ = O(‖∆p‖2) and‖F (z[1](p), p) − F (z(p), p)‖ = O(‖∆p‖2). Therefore Buskens [2] proposes a corrector iteration method, which usesartificial deviations p in the constraints as new perturbation parameters G(z, p) − p = 0, to improve z[1](p):

z[2] := z[1] − dz

dp(p0)G(z[1](p), p). (5)

By repeating this step iteratively, the approximation converges towards an admissible solution z[∞](p) ensuringG(z[∞](p), p) = 0 while the objective improves to ‖F (z[∞](p), p) − F (z(p), p)‖ = O(||∆p||3).Hereafter real-time optimal control results are presented for the objective, cf. Knauer [3],

F (q, u) =

tf∫0

3∑i=1

(qi(t)2 + βiui(t)2)dt, βi > 0. (6)

Typically small initial value perturbations, cf. Figure 2, need only four corrector iteration steps to lower theconstraints’ violation under 10−6. For larger initial value perturbations, cf. Figure 3, about ten steps are needed.Computational times still keep reasonable with less than 0.01 s for each iteration on a Pentium III with 500MHz and81 discrete points of time. In Figure 3 real-time optimal solutions are depicted. The white trajectory indicates theunperturbed (nominal) solution, the exact perturbed solution is red, while the real-time approximation is marked bya dashed blue line. The nominal objective is F ∗ = 5.2381. The value of the objective Freal−time = 5.5249 calculatedin real-time corresponds very good with the exact perturbed solution Fexact = 5.5201.

Figure 1: The degrees offreedom

Figure 2: Real-time optimal solution forsmall perturbations

Figure 3: Real-time optimal solution forlarge perturbations

5. References

1 C. Buskens: Optimization Methods and Sensitivity Analysis for Optimal Control Problems with Control and StateConstraints. (in German). Ph.D. Thesis. University of Munster (1998).

2 C. Buskens: Real-Time Optimization and Real-Time Optimal Control of Parameter-Perturbed Problems. (in German).Habilitation Thesis. University of Bayreuth (2002).

3 M. Knauer: Sensitivity Analysis and Different Objective Functions for Optimal Path Planning of Industrial Robots. (inGerman). Diploma Thesis. University of Bayreuth (2001).

Matthias Knauer, Christof Buskens, Universitat Bayreuth, 95440 Bayreuth, Germany

Section 20: Optimization 516