A manipulator is a series of linked segments connected by actuated joints that can manipulate objects without direct contact. The Jacobian inverse allows computation of the joint velocities required to achieve a desired end effector velocity as long as the Jacobian matrix is square and non-singular. The Jacobian inverse has applications in robotics such as handling materials, surgery, space and underwater exploration, and entertainment.

1. JACOBIAN INVERSE
MANIPULATOR
Hitesh Mohapatra
https://www.linkedin.com/in/hiteshmohapatra/

2. WHAT IS A MANIPULATOR?
■ A manipulator is a series of links, which connects the hand to the base, with each link
connected to the next by an actuated joint. If a coordinate frame is attached to each link,
the relationship between two links can be described with a homogeneous transformation
matrix.
■ In robotics a manipulator is a device used to manipulate materials without direct contact.
The applications were originally for dealing with radioactive or biohazardous materials,
using robotic arms, or they were used in inaccessible places.
■ It is an arm-like mechanism that consists of a series of segments, usually sliding or jointed
called cross-slides which grasp and move objects with a number of degrees of freedom.
■ Additionally, manipulator tooling gives the lift assist the ability to pitch, roll, or spin the part
for appropriate placement. An example would be removing a part from a press in the
horizontal and then pitching it up for vertical placement in a rack or rolling a part over for
exposing the back of the part.

3. WHY KINEMATICS?
■ Kinematics is the study of how things move.
■ It describes the motion of a hierarchical skeleton structure.
■ Forward Kinematics
Computation of the position, orientation and velocity of the end effector, given the
displacements and joint angles.
■ Inverse Kinematics
Computation of the joint displacements and angles from the end effectors position and
velocity.

4. MATHEMATICAL REPRESENTATION OF
KINEMATICS

5. Relationship between Forward Kinematics
and Inverse Kinematics
■ The configuration of a robot in terms of its joint angles is “Joint Space” and
■ The configuration of a robot with its end effector in space as it’s “Cartesian Space”
configuration.
■ We can use the concepts and mathematics to figure out the position of the end
effector in space, given that we know the joint angles, using a tool called JACOBIAN.

6. Two Degree of Freedom Planar Manipulator

7. TWO DEGREES OF FREEDOM PLANAR
JACOBIAN MANIPULATOR
■ We can then describe the end effector position, xₑ and yₑ, in space in terms of the
joint angles θ₁ and θ₂ as follows:
■ These two linear equations show that any change in either θ₁ or θ₂ will cause a
change in xₑ and yₑ.
■ This means the end effector position is dependent on the variables of the joint
angles in Jacobian manipulator.

8. ■ Taking the partial derivatives of the above equations, the differential relationship
between end-effector position and joint angle as such:
■ We can write the above equation in a much more compact format by stating:
dx = J . dq

9. DECODING THE EQUATION
■ The q vector is known as the system state.
■ The J matrix is referred to as the Jacobian matrix.
■ A Jacobian, mathematically, is just a matrix of partial differential equations.
■ Jacobian matrix of a manipulator describes the velocity of the system and how it
affects the end effector’s position.
■ The Jacobian for the system shown above can is:

10. JACOBIAN MATRICES
■ Jacobian matrix is a tool used throughout robotics and control theory.
■ Basically, a Jacobian defines the dynamic relationship between two different representations of a system.
■ For example, if we have a 2-link robotic arm, there are two obvious ways to describe its current position:
■ 1) the end-effector position and orientation and
■ 2) as the set of joint angles .
■ The Jacobian for this system relates how movement of the elements of causes movement of the elements of
■ Formally, a Jacobian is a set of partial differential equations:
■ With a bit of manipulation we can get a neat result:
This tells us that the end-effector velocity is equal to the Jacobian, multiplied by the joint angle velocity.

11. Why is Jacobian important?
■ To control in operational (or task) space.
■ Task space is a different space than the one that we can control directly, used in
planning a trajectory.
■ In our robot arm, control is effected through a set of motors that apply torque to the
joint angles, BUT what we’d like is to plan our trajectory in terms of end-effector
position (and possibly orientation), generating control signals in terms of forces to
apply in (x,y,z) space.
■ Jacobians allow us a direct way to calculate what the control signal is in the space
that we control (torques), given a control signal in one we don’t (end-effector forces).
The above equivalence is a first step along the path to operational space control.

12. USING THE JACOBIAN TOOL
■ We can formulate the dynamics of the system as follows:
■ The Jacobian maps the relationship between joint velocities (q) and end effector velocities(𝑣𝑒).

13. BUILDING THE JACOBIAN
■ First, we need to define the relationship between the (x,y,z) position of the end-effector and the robot’s joint angles,
(q_0, q_1). However will we do it? Well, we know the distances from the shoulder to the elbow, and elbow to the
wrist, as well as the joint angles, and we’re interested in finding out where the end-effector is relative to a base
coordinate frame.
■ Recall that transformation matrices allow a given point to be transformed between different reference frames. In this
case, the position of the end-effector relative to the second joint of the robot arm is known, but where it is relative to
the base reference frame (the first joint reference frame in this case) is of interest. This means that only one
transformation matrix is needed, transforming from the reference frame attached to the second joint back to the
base.

14. ■ The translation part of the transformation matrices is a little different than before
because reference frame 1 changes as a function of the angle of the previous joint’s
angles. From trigonometry, given a vector of length r and an angle q the x position of
the end point is defined r ; cos(q), and the y position is r ; sin(q). The arm is
operating in the (x,y) plane, so the z position will always be 0.
■ Using this knowledge, the translation part of the transformation matrix is defined:

15. ■ which transforms a point from reference frame 1 (elbow joint) to reference frame 0
(shoulder joint / base).
■ The point of interest is the end-effector which is defined in reference frame 1 as a
function of joint angle, q_1 and the length of second arm segment, L_1:

16. Inverse Kinematics of Manipulators In Motion
• Mathematically, it can only have one solution, as long as the Jacobian is non-singular.
• This can cause the controller to tell the manipulator it needs to be at some impossibly high velocity,
and the controller will try and follow it, causing sometimes catastrophic results.

17. ■ By inverting the Jacobian matrix we can find the joint velocities required to achieve a
particular end-effector velocity, so long as the Jacobian is not singular.
■ The inverse Jacobian allows us to determine:
joint velocity given the end-effector velocity
■ In order to be inverted the Jacobian matrix must be square and non-singular.
■ In order to be square the dimensions of the robot's task and configuration spaces must
be equal.
■ In order to be non-singular, the robot's joint coordinates must avoid certain "singular"
configurations.

18. Applications of Jacobian Inverse
Manipulator in Robotics
The robotics has been instrumental in the various domains such as −
■ Industries − Robots are used for handling material, cutting, welding, color coating,
drilling, polishing, etc.
■ Military − Autonomous robots can reach inaccessible and hazardous zones during war. A
robot named Daksh, developed by Defense Research and Development Organization
(DRDO), is in function to destroy life-threatening objects safely.
■ Medicine − The robots are capable of carrying out hundreds of clinical tests
simultaneously, rehabilitating permanently disabled people, and performing complex
surgeries such as brain tumors.
■ Exploration − The robot rock climbers used for space exploration, underwater drones
used for ocean exploration are to name a few.
■ Entertainment − Disney’s engineers have created hundreds of robots for movie making.