UNIT II MOBILE ROBOT KINEMATICS

2.1 KINEMATIC MODELS AND CONSTRAINTS

Kinematics is the most basic study of how mechanical systems behave. In mobile
robotics, we need to understand the mechanical behavior of the robot both in order to design
appropriate mobile robots for tasks and to understand how to create control software for an
instance of mobile robot hardware.
Of course, mobile robots are not the first complex mechanical systems to require such
analysis. Robot manipulators have been the subject of intensive study for more than thirty years.
In some ways, manipulator robots have been much more complex than early mobile robots: a
standard welding robot may have five or more joints, whereas early mobile robots were simple
differential drive machines.
• Locomotion is the process of causing an autonomous robot to move.
– In order to produce motion, forces must be applied to the vehicle
• Dynamics – the study of motion in which these forces are modeled

– Includes the energies and speeds associated with these motions

• Kinematics – study of the mathematics of motion without considering the forces that
affect the motion.
– Deals with the geometric relationships that govern the system
– Deals with the relationship between control parameters and the behavior of a systemin state
space
 UNIT II MOBILE ROBOT KINEMATICS
Kinematics Model:

Representing Robot Position:

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Example:

Forward kinematic models:

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Wheel kinematic constraints:

Fixed Standard Wheel:

 UNIT II MOBILE ROBOT KINEMATICS

Example:

• Suppose that the wheel A is in position such that α = 0 and β = 0

• This would place the contact point of the wheel on XI with the plane of
• The wheel oriented parallel to YI. If θ = 0, then this sliding constraint reduces to:

(i) Steered standard wheel:

• No vertical axis of rotation → No steering
• A in polar coordinate
• The angle of wheel plane relative to chasis - B fixed
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(ii) Castor Wheel:
• Steer around a vertical axis
• Different vertical axis of rotation from contact point.
• Any motion orthogonal to the wheel plane must be balanced by and equivalent andopposite
amount of castor steering motion

(iii) Swedish Wheel

Standard+1 DOF
 UNIT II MOBILE ROBOT KINEMATICS
(iv) Spherical Wheel
• No direct constraints on motion.
• Has no principal axis of rotation so no appropriate rolling or sliding constraint exist.
• Omnidirectional
• No effects on robot chasis kinematics.
• The Eq. is similar to the fixed standard wheel but here the direction of movement isarbitrary

• Compute the kinematic constraints of a robot with M wheels.

• Combine the constraints that arise from all the wheels based on the placement of
them on the robot chassis.
• Only standard fixed and steering wheels have constraints.
Robot kinematic constraints
• Compute the kinematic constraints of a robot with M wheels.
• Combine the constraints that arise from all the wheels based on the placement of
them on the robot chassis.
• Only standard fixed and steering wheels have constraints.
• N wheels Nf + Ns.
The Rolling constraint:
It is the constraint that all standard wheels must spin around their horizontal axis an appropriate
amount based on their motions along the wheel plane so that rolling occurs at the ground contact
point.
 UNIT II MOBILE ROBOT KINEMATICS
The Sliding constraint:
The components of motion orthogonal to the wheel planes must be zero for allstandard
wheels.
Sliding constraint in standard wheels has the most significant impact on defining the
overall maneuverability of the robot chassis.
The combination of wheel rolling and sliding constraints describes the kinematic
behaviour.
Example: A differential-drive robot.
By defining alpha and beta angles for both wheels J1f and C1f matrices can be computed.

2.2 MOBILE ROBOT MANEUVERABILITY

• Kinematic mobility: Robots ability to directly move in the environment.
• The basic constraint in mobility is satisfying the sliding constraint.
Degree of Mobility:
For both of these constraints to be satisfied, the
motion vector R (θ)ξ1_dot must belong tonull space
of the projection matrix C1 (βs )
Instantaneous center of Rotation:
•Those equations can be represented
geometrically by ICR.
•Zero motion line. Perp to wheel plane
•ICR geometric construction demonstrates how
robot mob. is a function of the # of the
constraints not the # of wheels.
•Robot chasis kinematics is therefore a
function of the set of independent constraints.
Arising from all standard wheels.
UNIT II MOBILE ROBOT KINEMATICS
 UNIT II MOBILE ROBOT KINEMATICS

2.3 MOBILE ROBOT WORKSPACE

• How can a robot use its control degrees of freedom to position itself in the
environment?
• What are the possible trajectories that a robot can follow?
The answer is related to the robots Degrees of Freedom (DoF) and DifferentiableDegrees
of Freedom (DDoF)
Differentiable Degrees of Freedom (DDoF) affect the ability of the robot to achieve
various paths
Degrees of Freedom (DoF) affect the ability of the robot to achieve various poses
Differentiable Degrees of Freedom (DDoF)
DDoF = δm (degree of mobility)
Example:
Bicycle -> δM = δm + δs = 1 + 1 = 2
DDoF = 1 but DoF = 3
Omnibot -> δM = δm + δs = 3 + 0 = 2
DDoF = 3 and DoF = 3

DDOF ≤ δM ≤ DOF
UNIT II MOBILE ROBOT KINEMATICS
Holonomic robots:
• In mobile robotics, the term refers specifically to the kinematic constraints of the robotchassis
• A holonomic robot has zero non-holonomic kinematic constraints
• A holonomic kinematic constraint can be expressed as an explicit function of position
variables only.
• A non-holonomic kinematic constraint requires a differential relationship and it cannotbe
integrated to provide a constraint in terms of the position variables only
Example:
Let’s consider a bicycle with a locked front wheel
δM = 1
and
[–sin(α+β)cos(α+β) l cosβ ] R(θ)ξl + rϕ· = 0
which can be replaced by
ϕ = (x ⁄ r) + ϕ
therefore this bicycle is holonomic!
A more intuitive way to describe holonomic robots is to say that:
DDoF = DoF
must hold.
In general we require DDoF = DoF = 3, meaning that we ’prefere’ omnidirectionalrobots
Path and trajectory:
Although we like holonomic robots, there are some serious considerations:
• Their design is more complex and expensive
• They are less stable during movement
Consider the Omnibot!

UNIT II MOBILE ROBOT KINEMATICS

2.4 Beyond Basic Kinematics
More things are to be considered in real life:
• Dynamic constraints due to speed and forces
• Violation of the previously defined kinematic models
• Presence of friction
• Actuation of the available degrees of freedom

2.5 MOTION CONTROL

Open loop control (trajectory following)
• Not always easy to find a feasible trajectorythat
meets the constraints
• Not smooth trajectories
• Not adaptive to changing environments

Disadvantages:
• It is not at all an easy task to pre-compute a feasible trajectory if all limitations and
constraints of the robots velocities and accelerations have to be considered.
• The robot will not automatically adapt or correct the trajectory if dynamic changes
• of the environment occur.
• The resulting trajectories are usually not smooth, because the transitions from one
• trajectory segment to another are for most of the commonly used segments (e.g. lines
• and part of circles) not smooth. This means there is a discontinuity in the robots
acceleration.
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Feedback control
• Define the kinematic model of the robot
• Find a control matrix K such that the robot moves to the desired position
• Use of K must result in a ’stable’ system
Example

Mobile robot model

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Closed loop system

Unique equilibrium point at

(ρ,α,β) = (0, 0,0)
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PART - A Questions
1. What are the kinematic constraints?
Kinematic constraints are equations that control the motion of solids, faces, edges,
or points. Add a Prescribed Displacement constraint to enter expressions for constraints. You can
define the equations using predefined coordinate systems as well as custom coordinate systems..

2. What is a kinematic model?

A kinematic model is a mathematical description of the robot: its functional dimensions
and DoF. It describes the robot's workspace, its positional capabilities and constraints. Most often
used for describing robot kinematics, which is also the case in this method, is the modified
Denavit–Hartenberg (MDH) notation.

3. What are the 3 concepts of kinematics?

There are three basic concepts in kinematics - speed, velocity and acceleration.

4. What are the parameters of kinematics?

Definition of the different kinematic parameters: (a) horizontal adduction of the shoulder;
(b) internal rotation of the shoulder; (c) shoulder abduction; (d) rotation of the pelvis and upper
torso; (e) finger flexion; (f) wrist flexion; (g) elbow flexion; (h) trunk tilt forwards and knee
extension;

5. What are the 5 major kinematic quantities?

The 5 major kinematic quantities are displacement (x-x0), time (t), initial velocity (v0), final
velocity (v), and constant acceleration (a). These quantities are commonly included when
describing the position and motion of an object.

6. What is the degree of mobility in mobile robot?

Degree of mobility(δm)
It is a measure of the # of DoF of robot chassis that can be immediatelymanipulated
through changes in wheel vel.

7. How a mobile robot can move forward or reverse?

Many mobile robots use a drive mechanism known as differential drive. It consists of 2
drive wheels mounted on a common axis, and each wheel can independently be driven either
forward or backward. By varying the velocities of the two wheels, we can vary the trajectories that
the robot takes.

8. What are the main characteristics of mobile robots?

Features of a mobile robot
• wireless communication.
• integrated safety.
• fleet simulation software.
• fleet management software.
• integration with the company's supervisory software.

9. What is the degree of maneuverability? What is the degree of maneuverability

dependent on?
Degree of maneuverability : overall degrees of freedom that a robot can. manipulate by
changing wheels' speed (direct mobility) and wheels' orientation. (indirect mobility).
Maneuverability depends on the position of the drive wheels.

10. What are 3 ways a robot can move?

There are three primary types of moves that a robot system uses to navigate around the
physical world: linear, joint, and circular moves.

11.What is the degree of freedom of wheeled robot?

Only robots that use exclusively wheels with three degrees-of freedom (3-DOF wheels)
will be able to freely move on a plane. This is because the pose of a robot on a plane is fully given
by its position (two values) and its orientation (one value).

11. How many degrees of freedom or flexible joints does a drone have?
Drones easily reach even the highest shelves in the warehouse. They are smaller than
drives, do not need any marks to navigate, and have 6 freedom degrees.

12. What is the workspace of the robot?

The workspace is a specification of the configurations that the end-effector of the robot
can reach, and has nothing to do with a particular task. For example, a planar robot with 2 revolute
joints, limited to ranges of motion of 180 and 150 degrees, has the workspace shown here.

13. What is mobile robot platform?

A mobile robotic platform is a machine with software, sensing elements and other
technologies that enable it to recognize, move and interact with the environment, depending on
possible unforeseen events.

14. How do I find the workspace of a robot?

For robot workspace, the vertices of the polygon consist of the boundary points. To
calculate the area of robot workspace is to compute the size of its corresponding polygon.
Supposed a polygon made up of line segments between N vertices (xi,yi), i=1 to N+1.

15. What is workspace analysis in robotics?

Robot working space is an important kinematic indicator. Exact computation of the
boundary shape and volume or area of the manipulator workspace is very important for its
optimum design and application. This paper describes the kinematics analysis of a 3-DOF
manipulator and the calculation of its workspace.
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PART - A Questions

16. What is kinematics in robotics?

Kinematics is the study of the relationship between a robot's joint coordinates and its
spatial layout, and is a fundamental and classical topic in robotics.

17. What is different types of kinematics in robotics?

Forward kinematics uses the kinematic equations of a robot to compute the position of the
end-effector from specified values for the joint parameters. The reverse process that computes the
joint parameters that achieve a specified position of the end- effector is known as inverse
kinematics.

18. What is motion control robotics?

Robot motion control enables articulated arms to move through the action of rotating and
sliding joints, and mobile robots to move through locomotion and steering. This controlled motion
enables these complex tasks with whatever end effector is appropriate on the robot.

19. What are the different types of motion control in robotics?

Typical motion control techniques are presented; i.e., servo control in joint space, servo
control in task space, dynamic control, and adaptive control. The servo control system is a control
system that makes the position and/or velocity of a mechanical motion system follow desired
values.
 Part-B Questions

1 Explain Kinematic models and constraints and itstypes?

2 Explain about mobile robot maneuverability?

3 Explain about mobile robot workspace?

4 Explain motion control in robotics?