## 3 Review of Trajectory Planning
### Approaches Based on Cartesian Motion Planning  
This approach is based on path planning (in task space) and the resolution of inverse kinematics (IK). Inverse kinematics is necessary to compute joint motion for exoskeletons. Cartesian coordinates (all points or knot points) are transformed into joint angles by using inverse kinematics. Inverse kinematics can be resolved using Jacobian-based methods [12], which exploit the available degrees of freedom (DOFs) of the robot's kinematic chain to achieve the desired end-effector pose. However, since the kinematic redundancy of the human arm allows for multiple arm configurations to complete the task in the three-dimensional workspace, it is difficult to obtain a nonunique solution for inverse kinematics in the human-robot workspace [25; 26]. To overcome this difficulty, an alternative approach based on swivel-angle computation is often used [26; 27]. In [25], the resolution of the redundancy through computation of the swivel angle made it possible to obtain unique solutions for the joint space. The application of trajectory planning based on path planning and inverse kinematics was presented in many studies [12; 28; 29]. In [28], the authors proposed a general kinematic model with Denavit-Hartenberg (DH) convention for the path generation of upper-limb exoskeleton robots. [PERSON] et al. used the reverse coordinate method to complete the inverse kinematics solutions; they also proposed a new multicubic polynomial-interpolation method for planning joint trajectories [12]. The Cartesian trajectory is commonly obtained by kinematic analysis [30], the Jacobian equation, geodesic curves [27; 31], DH convention, and other methods. Another way to define the Cartesian trajectory is by using motion models on the basis of the kind of task, such as circular motions [32] and periodic movements [33; 34]. These motion models can be derived from the results of the trajectory transformation corresponding to each joint movement during an ADL task. Accordingly, [PERSON] et al. proposed circular-arc trajectory planning for an upper-limb-exoskeleton rehabilitation robot [13]. For exoskeletons, once Cartesian trajectories and joint positions (through the inverse kinematics) are obtained, trajectory planning is completed in joint space using a polynomial-interpolation method. Although polynomial methods, of which so-called Bezier polynomials is the most popular, are a regular method for planning the desired trajectory of robotic systems, they require significant computing resources. For this reason, an alternative strategy based on the use of sigmoidal functions is proposed to construct reference trajectories [21]. However, such trajectories, which use mathematical expressions, are not exactly natural human trajectories since they are not related to the principles governing the control of human arm movements (such as optimization theory [35] and isochrony principle [32]). Thus, the trajectory can be planned in joint space by formulating a mathematical model of human motor behavior in order to plan the desired trajectories in a way similar to that of humans. Many mathematical models of human behavior, known as optimization-based methods, were proposed for humanlike motion planning [27; 36; 37; 38; 31]. These methods assume that human motions are generated by optimizing a known cost function. The most common of these models are the minimum-jerk [25; 32; 39], minimum-torque [40; 41], inertia-like [31; 42] and minimum-potential-energy [37] models. In [25], the spatial trajectory of the end effector was generated on the basis of a minimum-jerk model. A minimum-jerk approach was also adopted in [24; 39; 43; 44]. In [37], [PERSON] et al. exploited the minimization of potential energy. In the literature, the choice of the cost function to be optimized in typical approaches to generating human likeness in robotic motions is a hotly debated topic. In order to not have to choose a cost function, [PERSON] et al. exploited functional principal-component analysis (fPCA) for humanlike motion planning [42].