Advanced Mathematical Methods for Collaborative Robotics

The distinctive feature of human-robot collaboration is the absence of physical separation between robots and humans; i.e., they work with no safety fencing in the same workstations. This collaboration requires robots equipped with advanced sensors, such as force/torque sensors and vision systems. Moreover, advanced algorithms are needed to provide the robot with the level of intelligence required to accomplish the collaborative task at hand. In particular, two topics are especially relevant for human-robot collaboration: robot control using force feedback and robot guidance. Some literature review is presented below for these topics.


Introduction
The distinctive feature of human-robot collaboration is the absence of physical separation between robots and humans, i.e., they work with no safety fencing in the same workstations.
This collaboration requires robots equipped with advanced sensors, such as force/torque sensors, vision systems, etc. Moreover, advanced algorithms are needed to provide the robot the level of intelligence required to accomplish the collaborative task at hand. In particular, two topics are especially relevant for human-robot collaboration: robot control using force feedback, and robot guidance. Some literature review is presented below for these topics.
Robot control using force feedback. A position/force robot control is presented in [1] using fuzzy techniques for grinding applications. A force overshoot-free method was developed in [2] using impedance control for force tracking. A control technique based on the quarry matrix was proposed in [3] to control the tool pose and the force in the tool Z-axis.
Robot guidance. The forces/torques exerted by the operator are usually obtained with a force sensor mounted at the robot wrist and the measurements of the sensor are typically converted into robot motions using compliance control [4]; although other variants and methods can be found in the literature, see [5] [6] among others.

Control techniques
One of the fundamental requirements for the success of a human-robot collaboration task is the capacity of the robot system to interact with the human operator. The quantity that describes the state of interaction more effectively is the contact force at the robot's endeffector. High values of contact force are generally undesirable since they produce stress.
To analyze this interaction, it is worth considering the behavior of the system under a position control scheme when contact forces arise. Since these are naturally described in the operational space, it is convenient to refer to operational space control schemes.
The impedance control is convenient to analyze the interaction of a manipulator with the environment under the action of an inverse dynamics control in the operational space. The block scheme representing impedance control is reported in Fig. 1.
The impedance control, in absence of interaction or along the directions of free motion, is equivalent to an inverse dynamics position control. Therefore, for the selection of the impedance parameters, it should be considered the need of high values to reject disturbances due to model uncertainties and to the approximations into the inverse dynamics computation.
Such high factors increase proportionally the gain matrix but the interaction forces must be limited to avoid undesired oscillations. Description of an interaction task between the robot and its environment in terms of natural constraints and artificial constraints, expressed with reference to the constraint frame, suggests a control structure that uses the artificial constraints to specify the objectives of the control system so that, the desired values can be imposed only onto those variables not subject to natural constraints [7]. The control action should not affect those variables constrained by the surface to avoid conflicts between the control and interaction with environment that could lead to undesired oscillations. This type of control structure is called hybrid force/motion control. The definition of artificial constraints involves both force and position/velocity variables. The block scheme of a hybrid force/motion control law is shown in Fig. 2

Mathematical basis
The kinematics of the robot system is typically considered to properly perform a closed-loop control. In particular, the pose p and the configuration q of the robot are related as follows [8]: where f is the robot kinematic function. The first-and second-order robot kinematics result in: where matrix J represents the robot Jacobian. Most works tackling robot control assume the existence of low-level joint controllers in charge of achieving the joint accelerations commanded by the designed robot control. Obviously, the robot dynamic model should be considered to design this underlying robot control. Moreover, the bandwidth of the designed control should be slower than that of the low-level joint controllers for stability reasons.

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