# Modeling and Control of a DC Motor Coupled to a Non-Rigid Joint
## 1 Introduction  
DC (direct current) motors are widely used components as electrical actuators, in cars, trains and driving high power hoisting machines, milling machines, among others. A field in which DC motors are also widely used is robotics. Whether using these to change the angle of a joint of a robotic manipulator, to drive a vehicle's traction system, or even to rotate the sensor of a LiDAR (Light Detection and Ranging), there are numerous uses for these in robotics. Another example are the legged-robotic vehicles that can be found in [1]. However, they have a non-linear behavior that results from the fact that they cannot rotate at any supply voltage, always having a minimum rotation torque (starting torque) that prevents rotation until a when a certain voltage is reached, which is commonly denominated by the motor dead zone. To compensate for these, several control methods can be applied, such as [2; 3; 4], just to state a few. Worm gearboxes are typically used when compactness is an important factor, as well as an orthogonal redirectioning is required. Yet, one of the greatest advantage of worm gears is its unique self-locking characteristic. This means that the gear can only rotate by its input side, and cannot be actuated through the load side. As [PERSON] and [PERSON] [5] state, these features make worm gearboxes useful formultiple types of robotic vehicles and manipulators. However, any gearbox has a main non-linearity, which is it allows some movement of its output shaft without rotating its input, called backlash. So, using a DC motor with a worm gearbox is a solution that guarantees that, for example in a robotic manipulator, when the arm's joint reaches the desired angle it does not move until the next required change. It can also be used on vehicles with legged locomotion systems, which can benefit from this type of motors since the joints are blocked in the desired position. Regarding this kind of locomotion, another advantage is that the vehicle can stay immobilized, while guaranteeing the position of the leg, even when the power is down. To increase safety and robustness in the performance of industrial robots, which otherwise could easily break their joints, or hurt someone, the use of non-rigid joints is increasingly found in the literature and in the industry. These mechanisms contain elastic components (such as springs), which introduce a mechanical energy storage in the system, potentially increasing the maximum output power [6]. In [7], [PERSON] et al. have designed and implemented successfully a controller based on the analysis of the inverse dynamics of the non-rigid joint, and have applied a structured controller. The testbed mixes rigid and elastic joints on an interesting configuration. The usage of non-rigid joints for exoskeletons is also expanding, possibly due to the decrease in the size of the actuators. An example is the work presented in [8] for a variable viscoelastic joint system. Another interesting application example for non-rigid joints are deployable mechanisms, as the one presented by [PERSON] et al. [9], where the singularities of a system like the presented are analyzed and successfully avoided. The Quasi-Direct Drive is also an example of the technology applied to actuate an non-rigid joint, as presented in [10] for a quadruped. As it can be easily understood, all the mentioned research has a same primary step--the modeling phase--applied to mathematically translate the behavior of the system. Modeling the system is crucial to understand its behaviour under multiple conditions and also to develop its control. Having the model, one can put a signal in the process input and analyse how the system react, obtaining the system output. The system's transfer function can be interpreted as the process model. Robotic systems often include DC motors and having a realistic model is crucial to improve the control performance. This is required to be able to design suitable controllers, test them in simulation and obtain relevant results.