## 1 Introduction  
Trajectory planning is an ever-challenging issue in robotics. A typical category of trajectory planning problems is time-optimal trajectory planning [1, 2, 3]. Another popular category is to minimize energy consumption [4, 5, 6, 7, 8]. In addition, limiting the joint speed and acceleration [9] and minimizing the actuation force [10], joint torques [11], and joint wear [12] have been widely studied. However, most studies have been limitedly conducted under a predefined fixed path.  
To overcome the above limitation, model predictive control (MPC) has been recently used to solve time-optimal trajectory problems with given initial and end points only [13, 14]. This approach can create both the optimal path (i.e., the locus of points in the task space) and the optimal trajectory (i.e., a path on which a timing law is specified) for a minimal operation time. Note that this is essential for point-to-point motion control with no predefined path. However, implementing MPC requires an explicit mathematical model that can precisely express the dynamic performance of the system. Thus, there exists a technical hurdle to implementing MPC-based trajectory planning for redundant manipulators because it is difficult to determine their mathematical performance model.  
In robotics, redundant manipulators are widely used because they provide an increased level of dexterity by virtue of having more degrees of freedom (DOFs) in the joint space than those required to execute a given task. The additional DOFs in the joint space can be used to avoid obstacles [15; 16], joint limits [17], and collision with other robots [18], and also to minimize joint velocity and acceleration [19] and joint torque [20]. They also provide an advantage in performing maintenance and inspection tasks [21]. Among various redundant manipulators, the distributed actuation mechanism (DAM) can enhance the fingertip force through relocating a redundant actuation point of a slider along the link without changing the posture [22; 23]. However, such redundancy inevitably leads to difficulty in precisely evaluating its performance and thereby determining an optimal trajectory for a given task.  
It is interesting to note that the bioinspired variable gearing of the three-link DAM has been recently investigated to maximize the manipulation performance [24]. Because the positions of the six redundant sliders, which actuate three joints, function as a gear ratio, the DAM can achieve better performance than the power-equivalent joint actuation mechanism (JAM). A gradient-based optimization algorithm was implemented to precisely conduct maximum performance evaluation (MPE) under redundant DOFs.  
This paper proposes a novel time-optimal trajectory planning method for redundant manipulators by conducting MPC using the MPE results. First, the DAM-based three-revolute-joint manipulator (DAM-3R) was analyzed to obtain the MPE results using gradient-based optimization. The MPE results were used to construct an explicit performance model for the redundant DAM-3R as the upper bound of the manipulator performance at each end-effector position. Then, MPC was conducted to determine a time-optimal trajectory using the explicit performance model. For comparison, the JAM-based three-revolute-joint manipulator (JAM-3R), which is also a redundant manipulator, was constructed and analyzed to have equivalent power and geometric specifications.  
This study is organized as follows. In Section 2, based on a brief explanation on the concept of the MPE, the MPE results for the DAM-3R and JAM-3R are obtained using gradient-based optimization. Section 3 describes the MPC formulation to determine a time-optimal trajectory for the DAM-3R. The JAM-3R is compared to demonstrate the validity and significance of the proposed method. The conclusions follow in Section 4.