Mathematical Problems in Engineering

Volume 2015, Article ID 918301, 13 pages

http://dx.doi.org/10.1155/2015/918301

## An Optimization-Based Impedance Approach for Robot Force Regulation with Prescribed Force Limits

^{1}Facultad de Ingeniería, Universidad Veracruzana, Boulevard Adolfo Ruiz Cortines S/N, Costa Verde, 94294 Boca del Río, VER, Mexico^{2}Centro de Investigación y de Estudios Avanzados del IPN, Unidad Zacatenco, Departamento de Ingeniería Eléctrica, Sección de Mecatrónica, Avenida Instituto Politécnico Nacional No. 2508, 07360 San Pedro Zacatenco, México, DF, Mexico

Received 31 October 2014; Accepted 6 January 2015

Academic Editor: Luis Rodolfo Garcia Carrillo

Copyright © 2015 R. de J. Portillo-Vélez et al. 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

An optimization based approach for the regulation of excessive or insufficient forces at the end-effector level is introduced. The objective is to minimize the interaction force error at the robot end effector, while constraining undesired interaction forces. To that end, a dynamic optimization problem (DOP) is formulated considering a dynamic robot impedance model. Penalty functions are considered in the DOP to handle the constraints on the interaction force. The optimization problem is online solved through the gradient flow approach. Convergence properties are presented and the stability is drawn when the force limits are considered in the analysis. The effectiveness of our proposal is validated via experimental results for a robotic grasping task.

#### 1. Introduction

The shifting paradigm of robotics from industrial applications to field oriented and human-robot interaction tasks triggered the use of force regulation strategies as a fundamental tool to successfully perform tasks with physical interaction between the robot and the environment, [1, 2]. Nowadays, several applications of robot force control strategies are running in many fields of the human productive and everyday activities [3, 4].

Traditionally, to successfully perform physical robotic interaction tasks with the environment, three stages are considered. In the first stage, using only motion control, the robot moves freely towards the surface of interaction (e.g., a human tissue or an iron surface of an object). In the second stage, using a transition controller, contact between the robot and the surface is established with an impact caused by the transition from free motion to constrained motion. Finally, in the constrained motion stage, the goal for the robot is to keep contact with the environment by using a robot force regulation strategy. Robot force control has thoroughly studied the three stages of robot force interaction, focusing mainly in the third one [5, 6].

Two major approaches for robot force regulation are distinguished in the literature. Indirect force control (or impedance control) and direct (or explicit) force control [7]. Impedance control implicitly considers the three stages of interaction with the environment without a switching strategy from free motion to constrained motion as opposed to direct motion control approaches.

Despite the amount of work regarding robot force regulation, new problems arise in practical applications such as human-robot interaction, robotic surgery and industrial tasks for precision finishing of materials. For example, the notion of safety in human-robot interaction as well as kinematic and dynamic uncertainty in grasping tasks, lead to the definition of new constraints on the interaction force, which along with other constraints are recently focus of interest of control in robotics, [8]. The main restriction is that such problems must be online solved.

On limiting the interaction force in robotics, few approaches are considered in the literature. One approach is to design and implement mechanical devices which passively limit excessive forces when the robot interacts with the environment [9–11]. Another approach is to fully design variable stiffness compliant actuators which, together with special software development, which might render the required force safety levels for a given force controlled task [12, 13]. Both approaches, require a complete mechanical design of the force limiter device which demands a lot of design time and might result in a very expensive tool.

Alternatively, novel control schemes together with useful force sensors or parameter estimation techniques render an alternative solution to satisfy the required thresholds of undesired forces. In [14], an iterative feedback tuning algorithm is used to comply with the force safety requirements. This is achieved by formulating a constrained optimization problem and solving it using sequential quadratic programming. In [15], a neural network controller ensures that the control force remains within prescribed limits imposed by the user. This is achieved by properly adjusting the feedback gains of the controller. In [16], a force limiting adaptive controller is proposed to bound large interaction forces during a noncontact to contact transition with a mass-spring system. The semiglobal stability of the controller is given and experimental results without force sensors are presented. More recently, in [17], a force threshold-based position algorithm for legged locomotion is introduced. The algorithm uses preplanned motion and force measures to either elevate or depress the foot.

Our approach considers that, using the motion control loop of the robot, one can compute online modifications to the commanded robot trajectory using force sensor measurements, in order to satisfy prescribed force limits. To compute such trajectory modifications, a dynamic optimization problem for the force tracking error, considering prescribed interaction force limits, is formulated and solved.

The objective is to minimize the error for the interaction force. The optimization problem considers, as a performance index, the square of the weighted sum of the interaction force error as well as its time derivative [18]. In this work, it is assumed that the position of the end-effector of the robot is controlled by a classical position controller as a first control loop, see Figure 1. Moreover, a dynamic impedance model of the interaction between the robot and the environment is implemented via a classical position-based impedance approach as a second control loop [19, 20], see Figure 1. Such impedance model is an ordinary differential equation representing a dynamic equality constraint for the optimization problem, thus leading to a constrained DOP. As an inequality constraint for the DOP, interaction force limits are included, which are handled by penalty functions, [21]. The resulting constrained DOP is online solved through the gradient flow approach [22], which allows easily handling of the differential equation associated to the equality constraint (dynamic impedance model) for a robot. This is the main reason for using the gradient flow approach. The independent variable of the DOP is a computed reference trajectory arising from the impedance model for a given robot. As depicted in Figure 1, such reference trajectory () is online fed to the robot position controller (plus the desired impedance trajectory, ) in order to achieve the desired interaction force while accounting for undesired excessive or even insufficient forces. This represents the main difference with previous works and this feature is of great importance in some tasks such as robotic deburring in manufacturing processes and robotic grasping, where insufficient or excessive interaction forces might lead to break contact with the object or damage it, therefore failing the task.