RMV Mechanics

Mr. Anand H. D.
1
Department of Electronics & Communication Engineering
Dr. Ambedkar Institute of Technology
Bengaluru-56
Robotics & Machine Vision System: Sensors in Robotics
Robotics & Machine Vision System: Robot Mechanics

2
Introduction to manipulator kinematics
Homogeneous transformation and robot kinematics
Manipulator path control
Robot dynamics
Configuration of a robot controller
TOPICS TO BE COVERED

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Robot dynamics

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In order to develop a scheme for controlling the motion of a manipulator it is necessary to
develop techniques for representing the position of the arm at points in time.
We will define the robot manipulator using the two basic elements, joints and links.
Each joint represents 1 degree of freedom. The joints may involve either linear motion or
rotational motion between the adjacent links.
According to our definitions, the links are assumed to be the rigid structures that connect
the joints. Joints are labeled starting from 1 and moving towards the end effector with the
base being joint 1. Figure illustrates the labeling system for two different robot arms, each
possessing 2 degrees of freedom. .
By the joint notation scheme, the
manipulator in Fig.(a) has an RR
notation and the manipulator in
Fig. (b) has LL notation.

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Position Representation :
The kinematics of the RR robot are more difficult to analyze than the LL robot, and
we will make frequent use of this configuration (and extensions of it) throughout
this chapter.
Figure illustrates the geometric form of the RR manipulator. For the present discussion, our
analysis will be limited to the two-dimensional case. The position of the end of the arm may
be represented in a number of ways.
One way is to utilize the two joint angles θ1, and θ2. This is known as the representation in
joint space and we may define it as
Another way to define the arm position is
in “world” space. This involves the use of
a cartesian coordinate system that is
external to the robot.
The origin of the cartesian axis system is
often located in the robot's base.
The end-of-arm position would be defined
in world space as Pw=(x, y)

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This concept of a point definition in world space can readily be extended to three
dimensions, that is, Pw = (x, y, z).
Representing an arm's position in world space is useful when the robot must communicate
with other machines. These other machines may not have a detailed understanding of the
robot's kinematics and so a ‘neutral representation such as the world space must be used.
In order to use both representations, we must be able to transform from one to the other.
Going from joint space to world space is called the forward transformation or forward
kinematics while going from world space to joint space is called the reverse transformation
or inverse kinematics.
Forward Transformation of a 2-Degree of Freedom Arm :
We can determine the position of the end of the arm in world space by defining a vector for
link 1 and another for link 2.
Vector addition of (1) and (2) yields the coordinates x and y of the end of the arm (point
Pw) in world space
……………..(1)
……………..(2)
……………..(3)
……………..(4)

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Reverse Transformation of a 2-Degree of Freedom Arm :
 One approach is that employed in the control
system of the Unimate PUMA robot. In the
PUMA's control language, VAL, there is a set of
commands called ABOVE and BELOW that
determines whether the elbow is to make an
angle θ2 that is greater than or less than zero,
as illustrated in Fig.
 In many cases it is more important to be able to derive the joint angles given the end of-
arm position in world space.
 The typical situation is where the robots controller must compute the joint angles
required to move its end-of-arm to a point in space defined by the point's coordinates.
 For the two-link manipulator we have developed, there are two possible configurations
for reaching the point (x, y), as shown in Fig. This is so because the relation between the
joint angles and the end effector coordinates involve ‘sine' and cosine' terms.
 Hence we can get two solutions when we solve the two equations as given in the section
above. Some strategy must be developed to select the appropriate configuration.

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For our example, let us assume the θ2 is
positive as shown in Fig.
Using the trigonometric identities,
cos (A + B) = cos A cos B -sin A sin R
sin (A + B) = sin A cos B + sin B cos A
we can rewrite Eqs. (3) and (4) as
Squaring both sides and adding the two equations
yields
……………..(4)
Defining α and β as in Fig. we get

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Using the trigonometric identity
we get
Knowing the link lengths L1 and L2 we are now
able to calculate the required joint angles to place
the arm at a position (x, y) in world space.
……………..(4)

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Adding Orientation: A 3-Degree of Freedom Arm in Two Dimensions
 Accordingly, we will incorporate a third degree of freedom into the previous
configuration to develop the RR:R manipulator shown in Fig.
 The arm we have been modeling is very simple; a two-jointed robot arm has little
practical value except for very simple tasks.
 Let us add to the manipulator a modest capability for orienting as well as positioning a
part or tool.
 This third degree of freedom will represent a wrist joint. The world space coordinates
for the wrist end would be.
 We can use the results that we have
already obtained for the 2-degree of
freedom manipulator to do the reverse
transformation for the 3-degree of freedom
arm.

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 When defining the position of the end of the arm we will use x, y, and Ψ. The
angle Ψ is the orientation angle for the wrist. Given these three values, we can
solve for the joint angles (θ1, θ2 and θ3) using
 Having determined the position of joint 3, the problem reduces to the case of
the 2-degree of freedom manipulator previously analyzed.

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 Figure illustrates the configuration of a
manipulator in three dimensions.
 The manipulator has 4 degrees of freedom:
joint 1 (type T joint) allows rotation about the
z axis; joint 2 (type R) allows rotation about
an axis that is perpendicular to the z axis;
joint 3 is a linear joint which is capable of
sliding over a certain range; and joint 4 is a
type R joint which allows rotation about an
axis that is parallel to the joint 2 axis. Thus,
we have a TRL: R manipulator.
A 4-Degree of Freedom Manipulator in Three Dimensions
 Let us define the angle of rotation of joint 1 to be the base rotation θ; the angle of rotation
of joint 2 will be called the elevation angle φ; the length of linear joint 3 will be called the
extension L (L represents a combination of links 2 and 3); and the angle that joint 4
makes with the x – y plane will be called the pitch angle Ψ. These features are shown in
Fig.

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 The position of the end of the wrist, P, defined in the world coordinate system for the robot,
is given by
 Given the specification of point P (x, y, z) and pitch angle Ψ, we can find any of the joint
positions relative to the world coordinate (x4, y4, z4), which is the position of joint 4, as in
example,

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Robot dynamics

15
HOMOGENEOUS TRANSFORMATIONS AND ROBOT KINEMATICS
 The approach used in the previous section becomes quite cumbersome when a
manipulator with many joints must be analyzed.
 Another, more general method for solving the kinematic equations of a robot arm makes
use of homogeneous transformations.
 We describe this technique in this section, assuming the reader has at least some
familiarity with the mathematics of vectors and matrices.
 Let us begin by defining the notation to be used. A point vector, v = ai + bj + ck can be
represented in three-dimensional space by the column matrix
where a = x/w, b = y/w, c = z/w, and w is a scaling
factor.
For example, any of the following matrices can be used to
represent the vector v= 25i + 10j + 20k. Vectors of the above form can be
used to define the end-of-arm
position for a robot manipulator
(If w=0, then the vector
represents direction only)

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 It is important to notice that homogenous transformation is not unique as the coordinates of a
point can be scaled simply by changing the scaling parameter.
 A vector can be translated or rotated in space by means of transformation. The transformation is
accomplished by a 4 x 4 matrix, H. For instance the vector v is transformed into a vector u by the
following matrix operation: u = Hv
 The transformation to accomplish a translation of a vector in space by a distance a in x direction, b
distance in y direction and c distance in z direction is given by
 A vector v= 25i + 10j + 20k, performs a translation by a distance of 8 in x direction, 5 in
the y direction and o in the z direction. The translation transformation would be

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 The translated vector would be
 Rotation of a vector about each of three axes by an angle θ can be accomplished by a
rotation transformations. About the x axis rotation transformation is
About the y axis is About the z axis is
Rot(K.θ)

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Eg. Rotate the vector v=5i + 3j + 8k by an angle 90 degree about the x axis.
 About the x axis rotation transformation is
 Therefore rotation transformation is given by
 Note:

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Eg. Rotate the vector v=3i + 2j + 7k by an angle 60 degree about the z axis of the reference
frame. It is then rotated by an angle 30 degree about the x axis of the reference frame.
Find the rotation transformation.

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Eg. Rotate the vector v=3i + 2j + 7k by an angle 60 degree about the z axis of the reference
frame. It is then again rotated by an angle 30 degree about the x axis of the rotated frame.
Find the rotation transformation.

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Eg. A vector v=2i + 5j + 3k by an angle 60 degree about the z axis and translated by 3, 4 and
5 units in x, y and y directions respectively. Find the vector with reference to reference
frame.
About the z axis is

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 The use of the o, a, and n terms as direction vectors to describe the orientation of the wrist is
sometimes difficult. Another technique for describing the orientation of the robot end effector
involves the use of roll, pitch, and yaw.
 These are the same terms defined in earlier as the three possible degrees of freedom associated
with the wrist motion.
A Discussion on Orientation:
 In our present discussion we can define these terms more
precisely as the angles that are associated with these degrees of
freedom.
 Specifically, roll, pitch, and yaw are, respectively, the angles of
rotation of the wrist assembly about the z, y, and x axes of the
end-of-arm connection to the wrist.
 The definitions are illustrated in Fig. The end-of-arm connection
constitutes a coordinate reference frame.
 Given a, o, and n, we can find the roll, pitch, and yaw for the
wrist mechanism using the following equations:

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Robot dynamics

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 In controlling the manipulator, we are not only interested in the endpoints reached by the
robot joints, but also in the path followed by the arm in traveling from one point to
another in the workspace.
 Motion Types:
• There are three common types of motion that a robot manipulator can make in
traveling from point to point.
• These are slew motion, joint-interpolated motion, and straight line motion.
• Slew motions represent the simplest type of motion. As the robot is commanded to travel
from point A to point B, each axis of the manipulator travels as quickly as possible from
its respective initial position to its required final position.
• Therefore, all axes begin moving at the same time, but each axis ends its motion in an
elapsed time that is proportional to the product of its distance moved and its top speed
(allowing for acceleration and deceleration).
• Slew motion generally results in unnecessary wear on the joints and often leads to
unanticipated results in terms of the path taken by the manipulator.

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 In this scheme the path shapes in space and time are written as functions of joint angles.
This has the added advantage that there is no continuous correspondence between joint
space and Cartesian space. Hence there is no problem of singularity avoidance etc.
 Assuming that we need to move the tool end effector from the present position to the final
position, it is required to find a smooth function of the joint angles.
 From Inverse kinematics the joint angles at the initial θi; and final position θf, can be
found. As shown in Fig. the initial and final angle can be connected by many different
functions.
Joint Space Schemes

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Robot dynamics

27
Robot dynamics
 Accurate control of the manipulator requires precise control of each joint. The control of
the joint depends on knowledge of the forces that will be acting on the joint and the
inertias reflected at the joint (the masses of the joints and links of the manipulator).
 While these forces and masses are relatively easy to determine for a single joint, it becomes
more difficult to determine them as the complexity of the manipulator increases.
 Static Analysis: let us begin by considering the torques required by the joints to produce
a force F at the tip of the robot arm as shown in Fig. By balancing the forces on each link
we get
and That is
 The torques are the vector cross-products of the forces and the
link vectors r as developed earlier so that
and
Therefore, given a force vector F = (Fr,
Fy), the torques required by each joint
to generate that force at the tip can be
computed.

28
Robot dynamics
 Gravity was ignored in the previous analysis. It can be included by adding the forces exerted by
gravity on each link to the force balance equations in the previous section.
 Figure shows the manipulator with the force vectors Fg1 and Fg2 acting at the center of each link,
where the links have masses m1, and m2.
 Balancing the forces due to gravity only we get
Compensating for Gravity
 As we did before, the cross-products of the link vectors r and the
forces give us the torque due to gravity on each joint so that
If the manipulator is acting under gravity loads then these
torques must be added to the joint torques to provide a
desired output force. The obvious point of this is that if the
arm can be constructed with as low mass as possible more
of the motor torque will be used to provide forces at the end
of the arm for doing useful work, rather than just
supporting the mass of the links.

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Robot dynamics
Robot Arm Dynamics
 The topic of robot dynamics is concerned with the analysis of the torques and forces due
to acceleration and deceleration.
 Torques experienced by the joints due to acceleration of the links, as well as forces
experienced by the links due to torques applied by the joints, are included within the
scope of dynamic analysis.
 Solving for the accelerations of the links is difficult due to a number of factors. For one,
the acceleration is dependent on the inertia of the arm. However, the inertia is dependent
on the configuration of the arm, and this is continually changing as the joints are moved.
 An additional factor that influences the inertia
is the mass of the payload and its position
with respect to the joints. This also changes
as the joints are moved. Figure shows the
two-link arm in the maximum and minimum
inertia configurations.

30
Robot dynamics
 The torques required to drive the robot arm are not only determined by the static and
dynamic forces described above; each joint must also react to the torques of other joints
in the manipulator, and the effects of these reactions must be included in the analysis.
 Also, if the arm moves at a relatively high speed, the centrifugal effects may be
significant enough to consider.
 The various torques applied to the two-jointed manipulator are illustrated in Figure. The
picture becomes substantially more complicated as the number of joints is increased.

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Robot dynamics
 There are many different approaches to computing the dynamics of a robot arm.
 In this section we will briefly discuss about the Newton-Euler method and the Euler-
Lagrangian formulation.
 The Newton-Euler method is a 'force balance approach while the Euler-Lagrangian
approach is an 'energy-based approach.
 In the Newton-Euler formulation we are required to first compute the linear and angular
accelerations of the mass centers of each link. Then the inertia forces and torques acting at
the centre of each link is calculated as:
Dynamics
 where Fi is the force and Ni is the torque (moment) acting at the mass centre of each
link “i”, vi and wi are the linear velocity and angular velocity of each link centre I
is the inertia tensor of each link.
 After computing all the forces and moments the torques acting at each joint is found by
taking the Z component of the torque ‘Ʈ’ applied by one link on the other.

32
Robot dynamics
In the Lagrangian formulation we derive the equation of motion using a scalar function
called the Lagrangian, which is the difference between the kinetic and potential energies of
each link. The scalar Lagrangian function is written as
where KE and PE are the kinetic and potential energies of the link. The kinetic energy of a
link is given as
L = KE - PE
While the potential energy of the link is given by
where 'g' is the gravity vector that normally points downwards hence the negative sign, and Pi
is the position of the centre of the link.
The equation of motion of the manipulator is then derived from by
where Ʈ is the vector of joint torques. Using the Lagrangian formulation the final equation of
motion takes the form.

33
Robot dynamics

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 The elements needed in the controller include:
joint servocontrollers, joint power amplifiers,
mathematical processor, executive processor,
program memory, and input device.
 The number of joint servocontrollers and joint
power amplifiers would correspond to the number
of joints in the manipulator.
 These elements might be organized in the robot
controller as shown in Fig.
 Motion commands are executed by the controller
from two possible sources: operator input or
program memory.
 Either an operator inputs commands to the
system using an input device such as a teach
pendant or a CRT terminal, or the commands are
downloaded to the system from program memory
under control of the executive processor

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 In the second case, the set of commands have been
previously programmed into memory using the
operator input device(s).
 For each motion command, the executive processor
informs the mathematical processor of the
coordinate transformation calculations that must
be made.
 When the transformation computations are
completed, the executive processor downloads the
results to the joint controllers as position
commands.
 Each joint controller then drives its corresponding
joint actuator by means of the power amplifier.

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 Microprocessors are typically utilized in several of the
components of a modern robot controller.
 These components include the mathematical processor,
the executive processor, the servocontrollers, and the
input device. Each of the control boards makes use of a
common data buss and address buss.
 The microprocessors communicate with each other by
sending messages into common areas in the system
memory.
 This architecture provides several advantages in the
design of the controller. These advantages include
commonality of components, expansion of the system
to more joints, and information flow between joint
control elements.
 For example, the control configuration would permit
the sharing of feedback information among the various
joints, thus providing the opportunity to develop
algorithms for improving the individual joint dynamics.

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Reference
M.P. Groover, M. Weiss, R.N. Nagel, N.G. Odrey,
“Industrial Robotics-Technology, Programming
and Applications” Tata McGraw-Hill Education
Pvt Ltd,, 2008
For Further Studies

Prof. Anand H. D.
M. Tech.
Assistant Professor,
Department of Electronics & Communication Engineering
Dr. Ambedkar Institute of Technology, Bengaluru-56
Email: anandhdece@dr-ait.org
Phone: 9844518832

RMV Mechanics

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