Smart Robotics

Manipulator Differential Kinematics

Let's consider a manipulator. The direct kinematics equation can be written in the form:

Where:

• θ is the vector of joint variables.
• R is the rotation matrix describing the rotation of the end-effector, depends on θ.
• P is the position vector describing the position of the end-effector, depends on θ.

The goal of the differential kinematics is to find the relationship between the joint velocities and the end effector linear and angular velocities.

In other words, we want to express the end-effector linear velocity and angular velocity as a function of the joint velocities.

The relations we need are both linear in the joint velocities, which means that:

Where:

• Jv is a matrix relating the contribution of the joint velocities to the end-effector linear velocity V.
• Jω is a matrix relating the contribution of the joint velocities to the end-effector angular velocity ω.

In compact form, we obtain the manipulator differential kinematics equation:

very important because:

Singularities represent configurations in which the mobility of the structure is reduced (ex. it is not possible to impose an arbitrary motion to the end-effector).

When the structure is in a singular configuration, the inverse kinematics problem may have infinite solutions.

In the neighborhood of a singularity, small velocities in the operational space may cause large velocities in the joint space, so we cannot control properly the motion of the robot.

Singularities can be classified into two cases:

Boundary singularities: occur when the manipulator is either outstretched or retracted. These singularities do not represent a true drawback, since they can be avoided by simply not driving the manipulator to the boundaries of its reachable workspace.

Internal singularities: occur inside the reachable workspace and are generally caused by the alignment of two or more axes of motion, or else by obtaining particular end-effector configurations.

These singularities

• Translational velocity: the position vector v, that has 3 components.
• Rotational velocity: the vector containing the Euler angles θ, that has 3 components.

The difference between the analytical and the geometrical Jacobian is only the rotational part (since the angular velocities are typically different from the time derivative of the rotational velocity of the end-effector).

In compact form, we obtain the manipulator differential kinematics equation as the time derivative of the direct kinematics equation:

Where the analytical Jacobian is: Smart Robotics Pagina 28

Where the analytical Jacobian is:

It is different from the geometric Jacobian since the end-effector angular velocity wrt the base frame is not given by .

It is possible to find the relationship between the angular velocity and the rotational velocity for a given set of orientation angles.

Inverse Differential Kinematics

Inverse differential kinematics: is the problem of computing the velocity in the joint space given the end-effector linear and angular velocity in the operational space.

The solution of the inverse differential kinematics problem is the inversion of the Jacobian.

matrix, called manipulator inverse differential kinematics equation:

Depending on the number of DOF of the manipulator, if the Jacobian matrix is not square or full rank(so ), the inverse does not exist. In these cases it is possible to obtain an estimation of the joint velocities by exploiting suitable techniques (ex. pseudoinverse of the Jacobian matrix).

The configurations in which are singular configurations that are critical, and we need to avoid that by changing the planning trajectory.

Statics:

The problem of determining the relationship between the generalized forces applied to the end-effector and the generalized forces applied to the joints (forces for prismatic joints, torques for revolute joints) with the manipulator at an equilibrium configuration.

Let denote the vector of joint torques and the vector of the end-effector forces, where are the DOF and is the dimension of the operational space of interest.

We call torques the generalized forces at the joints, and forces the

generalized forces at the end-effector.

Principle of virtual work: the work performed by the external forces acting on a rigid body in equilibrium, due to any possible virtual displacement (spostamento), is always identically null.

The application of the principle of virtual work allows to determine the relationship between the end-effector forces and the joint torques, which is established by the transpose of the manipulator geometric Jacobian matrix: Smart Robotics Pagina 29

Trajectory Planning

lunedì 24 maggio 2021 17:34

Definitions

The goal of Trajectory Planning algorithm is to generate the reference inputs to the motion control system to ensure that the manipulator executes the planned trajectories.

The controller of the robot needs the trajectory as input, and we compute the torques applied to the joints to execute the planned trajectory.

In order to obtain the trajectory planning, we need to define the path and the trajectory to be implemented (ex. the time dependence of positions,

velocities and accelerations).The minimal requirement for a manipulator is the capability to move from an initial posture to a- final assigned posture.The transition should be characterized by motion laws requiring the actuators to exert joint- generalized forces which do not violate the saturation limits and do not excite the structure ofthe manipulator.Thus, it is necessary to use planning algorithms that generate suitably smooth trajectories.

Trajectory: path on which a timing law is specified (ex. in terms of velocities and/or accelerations ateach point).Trajectory types: .Motion trajectory- .Interaction trajectory-Path: locus of the points in the joint space, or in the operational space, that the manipulator has tofollow in the execution of the assigned motion; a path is a pure geometric description of motion.Typically, the path is expressed in a parametric form as:operational spacejoint spaceTiming law: describes the trend of and .Ex. the end-effector has to move from to , so

• Path description.
• Path constraints (ex. obstacle).
• Constraints imposed by manipulator dynamics (ex. saturation of the limits).

• End-effector trajectories in terms of a time sequence of the values obtained by position, velocity and acceleration.

1. The "task planner" plans the sequence of nodes that must be crossed in the operational space. It is easier to

1. We work in the operational space.

2. The nodes can be far from each other, so we need to interpolate the nodes in the operational space.

3. Thanks to interpolation, the operational geometric path is defined in terms of position and orientation in space.

4. Since the control action of the manipulator is performed in the joint space, the points are sampled and the inverse kinematics is performed to move from the operational to the joint space.

5. Thanks to inverse kinematics, we obtain the sequence of nodes in the joint space.

6. The nodes are interpolated in the joint space.

7. Thanks to interpolation, we can obtain the geometric path in the joint space:

Ex. we want to go from A to C, passing through B; we define by interpolating the nodes in the operational space; then we move to the joint space to actuate the joints thanks to inverse kinematics; finally, we obtain 3 trajectories, one for each joint:

After the computation of the geometric path, we need to obtain a trajectory defining a motion.

timinglaw: operational space

joint space

If , there is a natural parametrization with time.

The motion timing law has to be defined according to some criteria:

• It is chosen based on the specifics of the task (ex. the robot has to stop in one point, hs to- proceed at a constant speed, etc…)
• It is chosen on the basis of optimum criteria (ex. minimum time, minimum energy, etc…)
• The constraints are imposed by the actuators (max torque, max speed) or by the task (max- inetrial load/acceleration on the payload).

Trajectory Planning Classification

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Classification based on the points:

• Point-to-point motion: we only assign the initial and final points of the path.
• Motion through a sequence of points: we assign a finite sequence of points along the path.

Class