# Task-Based Motion Planning Using Optimal Redundancy for a Minimally Actuated Robotic Arm
## 1 Introduction  
Kinematic redundancy refers to the difference between the number of degrees of freedom (DOF) of a robotic manipulator and the DOF needed to execute a given task [1]. In confined spaces where traditional robots are generally unable to operate, the most common solution is to use hyper-redundant robots (with more than 6 DOF and 3 DOF for 3D and 2D working spaces, respectively). Their large number of DOF enables them to easily maneuver around obstacles, avoid singularities and joint limits, and minimize the energy consumption [2; 3]. However, this comes at the cost of a notoriously complex design for control and motion planning due to the high dimensionality of the problem [4]. Analytical methods usually require an extremely complicated characterization of the motion planning problem [5], which is impractical to apply, whereas numerical approaches struggle with expensive computational efforts stemming from the complexity level of the planning problem [6]. Although planning optimization for hyper redundant robots has been extensively researched in the last two decades, there have been no major breakthroughs in the field of motion planning for modular serial robots. Modular robotic arms are composed of a varying number of modules (links, joints, etc.) assembled together to achieve a given purpose [7; 8]. The ability to change the assembly of the modules provides control over the level of redundancy of the robot so it can be adapted to a specific environment or task [8]. Although most algorithms have been developed to optimize the motion planning problem of hyper-redundant manipulators, the optimization of the redundancy itself remains an open challenge. Minimizing a manipulator's redundancy level to fit a task enableseasier and more efficient motion planning. Moreover, as the redundancy level increases, the motion plan is subjected to larger mechanical uncertainties, due to the accumulating backlash and the mechanical freedom associated with the joints. Thus, \"minimal redundancy\" clearly has both mechanical and computational advantages. We define minimal redundancy as the minimal number of joints that need to be actuated to perform a certain task. More specifically, when given a manipulation task such as pick-and-place, and the robot's mechanical properties are known beforehand (such as joint limits, module dimensions, grabbing tolerance, etc. ), the goal is to find the minimal subset of joints that ensures the existence of a path to the target. Then, based on this particular subset, we deploy a heuristic search algorithm to find the path itself. In this paper, we use a planning algorithm on a planar modular manipulator to find a least-cost trajectory with mechanical constraints and obstacle avoidance, while optimizing the number and order of the modules needed for robot assembly. Although the algorithm can be easily applied to various manipulators consisting of \(N\) revolute joints (NR robots), as shown below, it was applied here on the recently developed minimally actuated serial robot (MASR) [9; 10], a reconfigurable robotic arm with passive (non-motorized) rotary joints, and a single mobile actuator that travels along the arm to rotate the passive joints. In addition to the modularity of the MASR, it differs from other NR manipulators in two respects, both derived from its unique design. Since the passive joints are externally actuated by a single actuator, only one joint at a time can be rotated. However, the mobility of the actuator allows the gripper, which is not fixed to the end of the manipulator, to travel to any point along the arm. This design makes it possible to manufacture a low-cost light-weight robotic arm that can be adapted for a variety of purposes, such as agricultural automation, surgical operations, or space tasks. This paper is organized as follows. Section 2 presents the key mechanical characteristics of the MASR that the algorithm uses as a constraint to achieve an optimal solution.