## 1 Introduction
Another popular approach to manage the contrary facts of nonlinear system behavior and expensive online calculations for solving nonlinear OCPs in real-time is first linearizing the system in some manner and then exploiting the advanced techniques developed for linear MPC. For example, for linear MPC, highly applicable real-time implementations are available as open source [17]. For this, a common approach is to use a linear time-varying (LTV) model to approximate the nonlinear system, see References [18; 19; 20; 21]. This approach is reasonable and sufficient for the considered low-cost hardware environment because of its easy implementation in the presence of the equations of motion in a symbolic manner. Moreover, none of the mentioned MPC approaches for robotic manipulators use feedback directly obtained by a vision-based measurement of the robot's pose to control the robot as considered in this work. Sensing is essential for controlling the robot on the one hand and monitoring the environment on the other hand. Encoders usually provide control feedback, but camera-based systems can realize both robot state sensing and monitoring of the environment. They can detect the positions of robot links, humans, and workpieces, leading to safer and faster workflows. In this area, vision-based sensing is becoming more important, which is the second aspect of this paper. For example, it allows automatic recognition of a workpiece's position and orientation, which therefore does not have to be positioned exactly before a robot processes it [22]. In References [23; 24], stereo camera systems are used to track a flying ball and catch it with a 6-DOF robot arm. The humanoid robot _[PERSON]_ is even capable to catch two balls at the same time with its two arms [25; 26; 27]. Vision-based sensing is also an opportunity for robots with elastic links, for which the end-effector position cannot be computed from encoder signals, that is, joint angles, only. In this paper, both MPC and a vision-based approach for gaining control feedback in real-time are combined for a serial manipulator in an experimental study. Deriving the predictive control law includes a structured step-by-step design procedure involving parameter identification and derivation of the forward and inverse dynamics. The modeling is simplified to a large extent by exploiting the advantage of having the equations of motion in a symbolical and numerical form derived by the research code Neweul-M2[3]. To handle the heavy online computational burden originating from the derived nonlinear MBS-model, a linear time-varying MPC (LTV MPC) scheme is developed based on linearizing the nonlinear system with respect to the desired trajectory and the a priori known feed-forward control torque. Instead of only using the angles measured by the built-in incremental encoders of the robot as feedback for the MPC, the robot's current pose is detected based on images from a low-cost industrial camera as well. For this purpose, the robot is equipped with markers, which can be detected utilizing color filtering during image processing. From the detected markers' image coordinates, the joint angles are calculated and fed back to the controller. The conducted experiments reveal the feasibility, advantages, and robustness of the proposed LTV MPC scheme when dealing with encoder feedback and show the effect of vision-based feedback on the MPC-based control loop. Several aspects are making up the novel contributions of this work: To begin with, an encoder- and vision-based MPC is realized on a low-cost industrial-standard system, where a standard laptop PC performs the computations and process control. The controller shows that implementing and especially tuning an MPC is easy if the model equations, that is, the equations of motion, are available in symbolic form. In contrast to common control techniques, like PD or PID, adequate results are obtained with MPC without massive parameter tuning.