Robot Jacobian

The time derivative of the kinematics equations yields the Jacobian of the robot, which relates the joint rates to the linear and angular velocity of the end-effector. The principle of virtual work shows that the Jacobian also provides a relationship between joint torques and the resultant force and torque applied by the end-effector. Singular configurations of the robot are identified by studying its Jacobian.

Velocity kinematics

The robot Jacobian results in a set of linear equations that relate the joint rates to the six-vector formed from the angular and linear velocity of the end-effector, known as a twist. Specifying the joint rates yields the end-effector twist directly.

The inverse velocity problem seeks the joint rates that provide a specified end-effector twist. This is solved by inverting the Jacobian matrix. It can happen that the robot is in a configuration where the Jacobian does not have an inverse. These are termed singular configurations of the robot.

Static force analysis

The principle of virtual work yields a set of linear equations that relate the resultant force-torque six vector, called a wrench, that acts on the end-effector to the joint torques of the robot. If the end-effector wrench is known, then a direct calculation yields the joint torques.

The inverse statics problem seeks the end-effector wrench associated with a given set of joint torques, and requires the inverse of the Jacobian matrix. As in the case of inverse velocity analysis, at singular configurations this problem cannot be solved. However, near singularities small actuator torques result in a large end-effector wrench. Thus near singularity configurations robots have large mechanical advantage.

Kinematic equations

A fundamental tool in robot kinematics is the kinematics equations of the kinematic chains that form the robot. These non-linear equations are used to map the joint parameters to the configuration of the robot system. Kinematics equations are also used inbiomechanics of the skeleton and computer animation of articulated characters.

Forward kinematics uses the kinematic equations of a robot to compute the position of the end-effector from specified values for the joint parameters.^[3] The reverse process that computes the joint parameters that achieve a specified position of the end-effector is known as inverse kinematics. The dimensions of the robot and its kinematics equations define the volume of space reachable by the robot, known as its workspace.

There are two broad classes of robots and associated kinematics equations serial manipulators and parallel manipulators. Other types of systems with specialized kinematics equations are air, land, and submersible mobile robots, hyper-redundant, or snake, robots andhumanoid robots.

Kinematics

Robot kinematics applies geometry to the study of the movement of multi-degree of freedom kinematic chains that form the structure of robotic systems.The emphasis on geometry means that the links of the robot are modeled as rigid bodies and its joints are assumed to provide pure rotation or translation.

Robot kinematics studies the relationship between the dimensions and connectivity of kinematic chains and the position, velocity and acceleration of each of the links in the robotic system, in order to plan and control movement and to compute actuator forces and torques. The relationship between mass and inertia properties, motion, and the associated forces and torques is studied as part of robot dynamics.

  

Basic Concepts

Torque:

Torque is a measure of how much a force acting on an object causes that object to rotate. The object rotates about an axis, which we will call the pivot point, and will label 'O'. We will call the force 'F'. The distance from the pivot point to the point where the force acts is called the moment arm, and is denoted by 'r'. Note that this distance, 'r', is also a vector, and points from the axis of rotation to the point where the force acts.
Torque = r . F , The angle of force ‘F’ is perpendicular to the moment arm ‘r’ .
Example: Imagine pushing a door to open it. The force of your push (F) causes the door to rotate about its hinges (the pivot point, O). How hard you need to push depends on the distance you are from the hinges (r) (and several other things, but let's ignore them now). The closer you are to the hinges (i.e. the smaller r is), the harder it is to push. This is what happens when you try to push open a door on the wrong side. The torque you created on the door is smaller than it would have been had you pushed the correct side (away from its hinges).

Battery:

In electronics, a battery or voltaic cell is a combination of one or more electrochemical cells which store chemica energy. These cells create a voltage difference between the terminals of the battery. When an external electrical circuit is connected to the battery, then the battery drives an electric current through the circuit and electrical work is done. A battery is a device that converts chemical energy directly to electrical energy.
Types
Lead-acid battery , Lithium-ion battery, Nickel metal hydride battery. 

Current and Voltage:
Voltage is the electric force that causes the free electrons to move from one atom to another.
Voltage is electric potential energy per unit charge, measured in joules per coulomb. It is often referred to as "electric potential", which then must be distinguished from electric potential energy by noting that the "potential" is a "per-unit-charge" quantity.
Just like water needs pressure to force it through a hose, electrical current needs some force to make it flow. A volt is the measure of electric pressure. Voltage is usually supplied by a battery or a generator.
The scientific symbol for voltage is the letter "E" dating back to the early days of electricity when it was called "Electromotive Force." Electricians and wiring books use the letter "V", for Volts.

Alternating Current (AC):

Alternating Current (AC) flows one way, then the other way, continually reversing direction.
An AC voltage is continually changing between positive (+) and negative (-).

The rate of changing direction is called the frequency of the AC and it is measured in hertz (Hz) which is the number of forwards-backwards cycles per second. Mains electricity in the India has a frequency of 50Hz. 

Direct Current (DC):
Direct Current (DC) always flows in the same direction, but it may increase and decrease.
A DC voltage is always positive (or always negative), but it may increase and decrease.

Electronic circuits normally from a battery or regulated power supply,

require a steady DC supply which is constant at one value or a smooth DC supply which has a small variation called ripple.

this is ideal for electronic circuits.