# Predictable Trajectory Planning of Industrial Robots with Constraints
## 1 Introduction  
There are growing interests in trajectory prediction. Trajectory prediction can be classified into two kinds: trajectory prediction for future time and trajectory prediction for future tasks. Future actions have been predicted for obstacle avoiding and trajectory re-planning [1; 2]. Appropriate trajectories for future tasks have been predicted by using machine learning [3; 4]. Fast motion planning has been obtained from experience. The transfer of previously optimized trajectories to a new situation cannot be made in the joint space. Little attention has been paid to Cartesian trajectory prediction that predicts the motion performed by the robot end-effector under joint velocity/acceleration constraints.  
Many practical tasks impose constraints on robot motions, such as machining, welding, and cutting and so on. RRT (rapidly-exploring random tree)/PRM (probabilistic roadmap)-based approaches were used to plan the path with end-effector constraints [5; 6; 7; 8; 9]. The Cartesian positions of the end-effector at the via-points were achieved, but the motions of the end-effector between via points are not predictable, in view of the nonlinear effects introduced by the direct kinematics.  
There are limits of joint positions, velocities, and accelerations. To obtain smooth motions, the joint constraints should be bounded. Joint constraints are often met by optimizing appropriate objective functions, such as keeping the joints close to their range centers [10], using the joint ranges in a weighted pseudo-inversion [11], or defining an infinity norm to be minimized at the velocity levels [12]. However, this method did not track the assigned paths. Inverse kinematics was used to enable joint velocity and acceleration not exceeding their limits [13], which was achieved by scaling the task time. However, these methods cannot meet joint constraints.  
Generally, there are three questions. (1) The motions of the end-effector between via points are not predictable in trajectory planning with end-effector constraints. (2) There is no guarantee that the end-effector will track the assigned paths in trajectory planning with joint constraints. (3) End-effector constraints and joint constraints have seldom been considered simultaneously in the previous studies. In this paper, we attempt to solve these problems.  
Geodesic is an important concept in Riemannian geometry and can be used in robot trajectory planning [14; 15] and machine tool interpolation [16]. By selecting suitable Riemannian metrics and local coordinates, the direct output of the proposed method is joint trajectories.  
In this paper, we present a predictable trajectory planning with constraints using geodesics. The end-effector position, velocities, and accelerations can be predicted without forward kinematics. The prediction of end-effector motion properties can be obtained before the calculation of the geodesic method. While tracking the constrained end-effector paths, the joint velocity/acceleration constraints can be met by scaling the motion time. The trajectory based on this method is predictable and has a light computational burden. Furthermore, the end-effector motions are of high precision.  
This paper is organized as follows. The predictable trajectory planning with constraints is presented in Section 2. Using null constraints and a virtual end-effector constraint, predictable linear and circular trajectory planning are described in Section 3. Experiments using the Effort robot are illustrated in Section 4. Finally, the conclusion is given.