MATLAB scripts to implement common control techniques on planar robots.
Thus far, these scripts apply for the following model of planar robot:
- One- to two-degrees-of-freedom planar robots, intended to model modular wearable lower-limbs exoskeleton devices.
- Due to hardware limitations, only the second joint is actuated with respect to 2-degrees-of-freedom model.
- Three-degrees-of-freedom models are still under development.
A robotic device moves around and the configuration (i.e. state) of its degrees-of-freedom (i.e. joints) is described by what is called state-vector or state-variables, denoted by q. The state of the robot varies as environment/external forces are applied to it, such as friction, gravity, as well as the force of its own actuators. Once all these effects are considered, a set of equations are obtained, usually one per degree-of-freedom. If one organizes the equation terms according to their physical meaning, as well as their dependence on q and its derivatives, therefore the robot dynamics are described by the following matrices for example (Siciliano and Khatib, 2008):
H: Inertia
C: Coriolis forces
Fc, Fv: Friction forces (Coulomb and viscous)
G: Gravity forces
Fext: External forces
Ta: Actuator forces
To simpler planar models, these matrices are intuitive and easy to calculate means of Newton's equilibrium equations or by the Lagrange method. The figure below shows the model of three planar robots with, respectively, one, two and three degrees-of-freedom. These planar robots are intended to represent the model of a lower-limbs exoskeleton.
Here, the environment with which the robot interacts is modeled as a lumped-element system consisting of a damping and a stiffness (Benv and Kenv).
The environments are divided into two types:
- Stiff environments, in the sense that the environment is something that, to a certain extent, holds its own shape and interacts with the robot through its end-effector. The interaction, thus, takes place at task-level, inevitably. The environment can be considered a holonomic constraint.
- Soft (or fluid) environments, in the sense that the environment surrounds the robot (i.e. it does not have a particular object shape), opposing to different degrees to the robot's movement. In this case, the interaction is rather assumed to occur at joint level only. The environment can be considered a nonholonomic constraint.
The dynamics of the force sensor are neglected. In case they are considered, they can be modeled as the environment (i.e. with lumped-elements, Bfs and Kfs) and placed in series between the robot end-effector and the environment.
Force sensors are usually modeled as pure-stiffness elements. The resulting interaction force, in the case a force sensor is present, and in the case Kenv >>> and Kfs, would be the force sensor stiffness (Kint) times the force sensor displacement.
Fint = Kfsxfs
which can be considered as extent as the force the environment exerts onto the robot.
To deal with the interactions with the surrounding environment, two control loops exist within the present architecture. The inner-loop controller (which runs at a faster rate as the outer loop) always deals with reference tracking, whatever that reference might be (position, force, etc.), whereas the outer control loop deals with the interactions that may occur with the environment, whether this environment is virtual or real.
In the absence of interaction, the outer loop is not active and the inner loop rules the robot movement.
In the case the robot interacts with the environment, both loops are active.
The reason d'être of the outer loop is to change the robot reference, which is input in the inner loop.
The inner controller is a Proportional-Derivative-Integrative (PID). There are instances in which it is simplified to simpler forms by tuning one of the gains to zero.
With respect to the admittance scheme, the inner controller is a position controller.
With respect to the impedance scheme, the inner controller is a force controller.
The outer controller models the interaction of the robot with the environment. It is either of impedance (Z) or admittance (Y) type (HOGAN, 1985).
Both schemes model the interaction in order to achieve the behavior of a second-order system described by a desired inertia Md, damping Bd and stiffness Kd.
With regards to the impedance controller, it generates a force output (F) as a displacement error (x) is sensed:
Z = Mdxs2 + Bdxs + Kx
With regards to the admittance controller, it generates a displacement (x) as a force error (F) is sensed:
Y = (Mdxs2 + Bdxs + Kdx)-1 = Z-1
It is said that impedance and admittance form a dual relationship.
Impedance and admittance schemes have their intrinsic advantages and disadvantages. The figure below, drawn based on (Ott et. al, 2010) provides a meaningful insight on their main difference.
Below the block diagrams for the admittance and the impedance schemes can be seen, respectively. Both refer to a linear system. In case rotational joints are present, one or more of the following operations must be considered in the block diagrams: forward kinematics, inverse kinematics, Jacobian (or its transpose/inverse).
In task space, the interaction of the robot with the environment is represented by linear quantities, i.e. forces and vertical/horizontal displacements. It is as if the robot was moving around and, suddenly, something blocked its end-effector, for instance a wall or any other obstacle.
Thus, the environment offers resistance to robot displacement over X and Y directions, but the robot can rotate its joints freely (as long as the X and Y position of the end-effector do not change).
In joint space, the interaction of the robot with the environment manifests as angular quantities, i.e. torques and angular displacements. It is as if the robot had the movement of its joints constrained to certain values (similarly to the human joints), or as if the joints themselves had some sort of damping within its bearings.
Thus, the environment offers resistance the angular displacement performed by the joints of the robot. This restriction may occur only to certain values of displacement (again, similarly to human joints) or over the whole range of values of the joint.
There is no right or wrong whatsoever. It depends solely on the application, the situation you are trying to model, as well as on the hardware available.
In case you are trying to model the interaction of a industrial robot (UR5 for instance) with the surrounding environment, so it does not hit walls blindly, and rather stops to prevent penetrating them, task space is more adequate.
Now imagine the case you are trying to model a robotic arm which interacts with a fluid: the robotic arm attached to a submarine, for instance. The interaction with the environment occurs at joint level, that is, the environment (water) generates resistive torques that prevent the robot from moving freely. In this case, the joint space approach is more adequate.
The same is true with respect not only to the interaction with respect to the environment (outer control loop) but also with respect to the inner-control loop as well.
If you want to follow a desired joint trajectory or torque reference, joint space is your way to go.
Whereas if you want to follow a desired trajectory in space (X,Y,Z) coordinates, task space is what suits better.
Performance restrictions, hardware limitations, etc. may also lead you to approach the problem one way or another. Usually, whenever force sensing is present, specially in task-space control (when you control the joint position but rather sense the force at the end-effector, non-collocation problem), higher framerates (at least 1kHz) are necessary to ensure stability.
Commonly, if you are using impedance in your outer-loop, there will be a virtual second-order system between the robot and the desired trajectory, which will generate a force/torque whenever the robot deviates from the reference. It is as if there were virtual springs and dampers attaching the robot to the trajectory.
Usually, if you are using admittance in your outer-loop, you are generating a trajectory correction for your reference whenever a force is sensed. By doing this you prevent that the robot penetrates further into something that is blocking its way, which can cause damage to whatever is blocking it as well as to the robot.
Siciliano, B.; Khatib, O. Springer Handbook of Robotics. 1st edition. Springer-Verlag Berlin Heidelberg, 2008. 1611 p.
Hogan, N. Impedance Control: An Approach to Manipulation: Parts I, II and III. Journal of Dynamic Systems, Measurement, and Control, v. 107, n. 1, p. 1–24,03 1985. ISSN 0022-0434.
C. Ott, R. Mukherjee and Y. Nakamura, "Unified Impedance and Admittance Control," 2010 IEEE International Conference on Robotics and Automation, Anchorage, AK, 2010, pp. 554-561, doi: 10.1109/ROBOT.2010.5509861.
Here I provide a list of supporting MATLAB scripts which were not written by me. However, I may have made a few changes and adaptations to their original scripts (very few I'd say) so I'd recommed people to always refer to the scripts on this repository rather than the ones from MATLAB page.
Brandon Kuczenski (2020). hline and vline (https://www.mathworks.com/matlabcentral/fileexchange/1039-hline-and-vline), MATLAB Central File Exchange. Retrieved August 22, 2020.