Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 348178, 21 pages

http://dx.doi.org/10.1155/2015/348178

## Embedded Optimal Control of Robot Manipulators with Passive Joints

School of Communication Engineering, Rey Juan Carlos University, Fuenlabrada, 28943 Madrid, Spain

Received 12 March 2015; Accepted 19 August 2015

Academic Editor: Sivaguru Ravindran

Copyright © 2015 Alberto Olivares and Ernesto Staffetti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper studies the optimal control problem for planar underactuated robot manipulators with two revolute joints and brakes at the unactuated joints in the presence of gravity. The presence of a brake at an unactuated joint gives rise to two operating modes for that joint: free and braked. As a consequence, there exist two operating modes for a robot manipulator with one unactuated joint and four operating modes for a manipulator with two unactuated joints. Since these systems can change dynamics, they can be regarded as switched dynamical systems. The optimal control problem for these systems is solved using the so-called embedding approach. The main advantages of this technique are that assumptions about the number of switches are not necessary, integer or binary variables do not have to be introduced to model switching decisions between modes, and the optimal switching times between modes are not unknowns of the optimal control problem. As a consequence, the resulting problem is a classical continuous optimal control problem. In this way, a general method for the solution of optimal control problems for switched dynamical systems is obtained. It is shown in this paper that it can deal with nonintegrable differential constraints.

#### 1. Introduction

In underactuated manipulators the number of available control inputs is strictly less than the number of the degrees of freedom of the robot. However, the control problem for different underactuated manipulators may have different levels of difficulty.

This paper studies the optimal control problem for planar underactuated robot manipulators with two revolute joints that move in a vertical plane and, therefore, are subject to the gravity force. The presence of two revolute joints in the mechanical system will be denoted by. Both possible models of planar underactuatedrobot manipulators under gravity are considered, namely, the model in which only the shoulder joint is actuated, denoted by, which is usually called Pendubot, and the model in which only the elbow joint is actuated, denoted by, which is usually called Acrobot. The mechanical system obtained by removing both motors from therobot manipulator under gravity is also considered, which is actually a double pendulum and will be denoted by.

Underactuated robots are mechanical systems with second order nonholonomic constraints, because the dynamic equation of the unactuated part of the mechanical system is a second order differential constraint which, in general, is nonintegrable. In the absence of gravity it is not possible to integrate even partially this second order differential constraint in the dynamic model of therobot manipulator. However, in the presence of this second order nonholonomic constraint the system is controllable. On the contrary, in the dynamic model of therobot manipulator without gravity this differential constraint is completely integrable. It can be converted into a holonomic constraint that makes the system not controllable. As a consequence, the trajectory planning problem has a solution only for particular initial and final states. In the presence of gravity, the dynamic equation of the unactuated part of the mechanical system is a second order nonintegrable differential constraint for both the Pendubot and the Acrobot models and, therefore, both systems are controllable.

A comprehensive review of the control properties of underactuated robot manipulators can be found in [1], [2, Chapter 42].

In [3] the chaotic features of the double pendulum are studied by means of numerical methods. It is shown that, depending on the initial conditions, this system has a periodic, quasiperiodic, and chaotic behaviour, in which the motion of the system is unpredictable and small differences in initial conditions induce very different trajectories.

For both the Pendubot and the Acrobot models, the control objective is, in general, to drive the manipulator away from the stable equilibrium state and steer it at an unstable equilibrium state. Since both theand theunderactuated robot manipulators are linearly controllable, it is not difficult to control them around an equilibrium point. However, due to the gravitational drift, the region of the state space where the robot can be kept in equilibrium is reduced and consists of two disjoint manifolds. Moving between these two manifolds requires appropriate swing-up maneuvers, whose synthesis has been tackled by several control [4] and optimal control [5] techniques.

The underactuated robot manipulators studied in this paper are supposed to be equipped with brakes at the unactuated joints. The Pendubot and the Acrobot with brakes at the unactuated joints will be denoted byand, respectively, whereas themechanical system with brakes at both unactuated joints will be denoted by. This mechanical system can be regarded as a brake-actuated mechanical system and will be referred to as Brakebot. Thus, since each passive joint has two operating modes, unbraked and braked, there are two operating modes for the Acrobot and the Pendubot and four operating modes for the Brakebot.

Since the Brakebot is a passive system it can only move from higher to lower energy states and, therefore, the control objective is, in general, to drive the manipulator away from an unstable equilibrium state and steer it at a stable equilibrium state with appropriate swing-down maneuvers eliminating periodic, quasiperiodic, or chaotic behaviour. For the synthesis of these maneuvers, optimal control techniques can be employed. The cost functional to be optimized could be, for instance, the duration of the maneuver, the number of switches of the dynamical system between unbraked and braked modes, or the total duration of the evolution of the system in braked mode.

The problem studied in this paper can be stated as follows: given an initial state and a final state, find the sequences of modes, the corresponding trajectory, and available control inputs that satisfy the dynamic equation of the robot manipulator and steer the system between the initial and the final states optimizing a cost functional during the motion. Since the optimal sequence of modes has to be determined, this problem is actually an optimal control problem of a switched dynamical system.

Switched systems are particular hybrid systems [6] whose continuous state does not exhibit jumps at the switching instants. Solving the optimal control problem for a switched system implies finding the optimal sequence of switching instants, the optimal sequence of discrete modes, and the optimal value for the continuous control input.

The optimal control problem of switched dynamical systems can be formulated using integer or binary variables to model the choice of the optimal mode sequence. Optimal control problems involving binary or integer variables are called Mixed-Integer Optimal Control (MIOC) problems [7]. In [8], recent progress in theoretical bounds, reformulations, and algorithms for this class of problems have been reviewed and a comprehensive algorithm, based on the solution of a sequence of purely continuous problems and simulations, has been proposed. A hybrid optimal control technique based on MIOC has been presented in [9] and applied to several mechanical systems including a Pendubot with a brake at the unactuated joint.

Hybrid control techniques have been applied to both the Pendubot and the Acrobat models in [10] and [11], respectively.

In [12], the optimal control problem of switched systems has been solved via embedding into a continuous optimal control problem. It has been shown that, for quite a general class of optimal control problems for switched systems, the computational complexity of the Embedded Optimal Control (EOC) method is no greater than that of smooth optimal control problems. With this technique the switched system is embedded into a larger set of systems and the optimal control problem is formulated in the latter. The main advantages of this method are that assumptions about the number of switches are not necessary, integer or binary variables do not have to be introduced to model switching decisions between modes, and the optimal switching times between modes are not unknowns of the optimal control problem. As a consequence, the resulting problem is a classical continuous optimal control problem. This greatly reduces the computation time to find a solution with respect to the MIOC technique. In [13] the EOC approach introduced in [12] has been extended to incorporate hybrid behaviour due to autonomous switches and to systems with an arbitrary number of modes. In [14] the EOC method has been compared with other optimal control methods for switched systems over several dynamic models with autonomous and controlled switches. A comprehensive treatment of all the theoretical results obtained with the EOC approach to the optimal control problem of switched mechanical systems can be found in [15, Chapter 31].

In this paper, the optimal control of the Pendubot, Acrobot, and Brakebot has been studied using the EOC approach. The obtained results have been compared with those obtained using a Multiphase Mixed-Integer Optimal Control (MMIOC) method. Due to the nature of the problem, only controlled switches have been considered. The results have shown that the EOC approach is effective in solving optimal control problems for these underactuated mechanical systems, which are highly nonlinear and have nonholonomic constraints.

This paper is organized as follows. In Section 2 the dynamic models of the Pendubot, Acrobot, and Brakebot are described and in Section 3 their control properties are discussed. The optimal control problem for these dynamic systems is stated in Section 4 and in Section 5 the EOC approach is introduced. In Section 6 the results of the application of this method to several instances of optimal control problem are reported. Finally, Section 7 contains the conclusions.

#### 2. Dynamic Model of Underactuated Manipulators

The general dynamic model of a robot manipulator is described by the following second order differential equation:where the first term of this equation,, represents the inertial forces due to acceleration at the joints and the second term,, represents the Coriolis and centrifugal forces. The third term represents both the effects of friction at the joints and the effects of friction induced by electromagnetic friction brakes at the unactuated joints. The termrepresents the potential forces such as elasticity and gravity. The matrixon the right-hand side maps the external forces/torquesto forces/torques at the joints. Finally,represent the forces/torques at the joints, which are the control variables of the system. Explicit time dependence is omitted in this section to simplify the notation.

The links are supposed to be rigid, as well as the transmission elements. No external forces are supposed to be acting on the mechanical system. Under these hypotheses, the dynamic model of the robotic system reduces to

Only the effects of friction induced by the electromagnetic friction brakes at the unactuated joints will be taken into account. This means thatrepresents the braking torques induced by electromagnetic friction brakes at the unactuated joints.

A planar verticalmanipulator is composed of two homogeneous links and two revolute joints moving in a vertical plane, as shown in Figure 1, whereis the length of link,is the distance between jointand the mass center of link,is the mass of link, andis the barycentric inertia with respect to a vertical axis,, of link, for. In this case the two matricesandand the vectorhave the form whereis the vector of configuration variables,being the angular position of link 1 with respect to the -axis of the reference frameandthe angular position of link 2 with respect to link 1 as illustrated in Figure 1. The vectoris the vector of angular velocities andis the vector of angular accelerations. The vectoris the vector of control inputs, in whichis the torque applied by the actuator at Joint, andis the torque applied by the actuator at Joint. Constant [m/s^{2}] is the gravity acceleration. The parameters,, andin (3) and (4) have the following expressions,, and.