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## Kinematics

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### Kinematics

### Next Course

Amirkabir University of TechnologyComputer Engineering & Information Technology Department

Time to Derive Kinematics Model of the Robotic Arm

Kinematics of Manipulators

Objective:

- To drive a method to compute the position and orientation of the manipulator’s end-effector relative to the base of the manipulator as a function of the joint variables.

Degrees of Freedom

The degrees of freedom of a rigid body is defined as the number of independent movements it has.

The number of :

- Independent position variables needed to locate all parts of the mechanism,
- Different ways in which a robot arm can move,
- Joints

Degrees of Freedom

3 position

As DOF

3D Space = 6 DOF

3 orientation

In robotics:

DOF = number of independently driven joints

positioning accuracy

computational complexity

cost

flexibility

power transmission is more difficult

A manipulator may be thought of as a set of bodies (links) connected in a chain by joints.

- In open kinematics chains (i.e. Industrial Manipulators):

{No of D.O.F. = No of Joints}

Lower Pair

- The connection between a pair of bodies when the relative motion is characterized by two surfaces sliding over one another

Higher Pair

- A higher pair joint is one which contact occurs only at isolated points or along a line segments

Revolute Joint

1 DOF ( Variable - q)

Spherical Joint

3 DOF ( Variables - q1, q2, q3)

Prismatic Joint

1 DOF (linear) (Variables - d)

Robot Specifications

Number of axes

- Major axes, (1-3) => position the wrist
- Minor axes, (4-6) => orient the tool
- Redundant, (7-n) => reaching around obstacles, avoiding undesirable configuration

2

3

1

4

The PUMA 560 hasSIXrevolute joints.

A revolute joint has ONE degree of freedom ( 1 DOF) that is defined by its angle.

There are two more joints on the end-effector (the gripper)

Without loss of generality, we will consider only manipulators which have joints with a single degree of freedom.

A joint having n degrees of freedom can be modeled as n joints of one degree of freedom connected with n-1 links of zero length.

The Kinematics Function of a Link

The kinematics function of a link is to maintain a fixed relationship between the two joint axes it supports.

This relationship can be described with two parameters: the link length a, the link twist a

Is measured along a line which is mutually perpendicular to both axes.

The mutually perpendicular always exists and is unique except when both axes are parallel.

Project both axes i-1 and i onto the plane whose normal is the mutually perpendicular line, and measure the angle between them

Right-hand sense

Joint Parameters

A joint axis is established at the connection of two links. This joint will have two normals connected to it one for each of the links.

- The relative position of two links is called link offsetdn whish is the distance between the links (the displacement, along the joint axes between the links).
- The joint angleqn between the normals is measured in a plane normal to the joint axis.

Link and Joint Parameters

4 parameters are associated with each link. You can align the two axis using these parameters.

- Link parameters:

a0 the length of the link.

anthe twist angle between the joint axes.

- Joint parameters:

qn the angle between the links.

dn the distance between the links

Link Connection Description:

For Revolute Joints: a, , and d.

are all fixed, then “i”is the.

Joint Variable.

For Prismatic Joints: a, , and .

are all fixed, then “di” is the.

Joint Variable.

These four parameters: (Link-Length ai-1), (Link-Twist i-1(, (Link-Offset di), (Joint-Angle i)are known as theDenavit-HartenbergLink Parameters.

3

1

0

A 3-DOF Manipulator Arm

Links Numbering Convention

Base of the arm: Link-0

1st moving link: Link-1

. .

. .

. .

Last moving link Link-n

Link 2

Link 3

Link 1

Link 0

First and Last Links in the Chain

- a0= an=0.0
- a0= an=0.0
- If joint 1 is revolute: d0= 0 and q1 is arbitrary
- If joint 1 is prismatic: d0= arbitraryand q1 = 0

Affixing Frames to Links

In order to describe the location of each link relative to its neighbors we define a frame attached to each link.

- The Z axis is coincident with the joint axis i.
- The origin of frame is located where ai perpendicular intersects the joint i axis.
- The X axis points along ai( from i to i+1).
- If ai = 0 (i.E. The axes intersect) then Xiis perpendicular to axes i and i+1.
- The Y axis is formed by right hand rule.

Affixing Frames to Links

First and last links

- Base frame (0) is arbitrary
- Make life easy
- Coincides with frame {1} when joint parameter is 0
- Frame {n} (last link)
- Revolute joint n:
- Xn= Xn-1 when qn = 0
- Origin {n} such that dn=0
- Prismatic joint n:
- Xn such that qn = 0
- Origin {n} at intersection of joint axis n and Xnwhen dn=0

Affixing Frames to Links

Note: assign link frames so as to cause as many link parameters as possible to become zero!

The reference vector z of a link-frame is always on a joint axis.

The parameter di is algebraic and may be negative. It is constant if joint i is revolute and variable when joint i is prismatic.

The parameter ai is always constant and positive.

a i is always chosen positive with the smallest possible magnitude.

The Kinematics Model

The robot can now be kinematically modeled by using the link transforms ie:

Where

0nT is the pose of the end-effector relative to base;

Tiis the link transform for the ith joint;

and

nis the number of links.

The Denavit-Hartenberg (D-H) Representation

- In the robotics literature, the Denavit-Hartenberg (D-H) representation has been used, almost universally, to derive the kinematic description of robotic manipulators.

The Denavit-Hartenberg (D-H) Representation

- The appeal of the D-H representation lies in its algorithmic approach.
- The method begins with a systematic approach to assigning and labeling an orthonormal (x,y,z) coordinate system to each robot joint. It is then possible to relate one joint to the next and ultimately to assemble a complete representation of a robot's geometry.

The Link Parameters

ai = the distance from zi to zi+1.

measured along xi.

ai = the angle between zi and zi+1.

measured about xi.

di = the distance from xi-1 to xi.

measured along zi.

qi = the angle between xi-1 to xi.

measured about zi

General Transformation Between Two Bodies

In D-H convention, a general transformation between two bodies is defined as the product of four basic transformations:

- A translation along the initial z axis by d,
- A rotation about the initial z axis by q,
- A translation along the new x axis by a, and.
- A rotation about the new x axis by a.

A General Transformation in D-H Convention

D-H transformation for adjacent coordinate frames:

- D1. Establish the base coordinate system. Establish a right-handed orthonormal coordinate system at the supporting base with axis lying along the axis of motion of joint 1.
- D2. Initialize and loop Steps D3 to D6 for I=1,2,….n-1
- D3. Establish joint axis. Align the Zi with the axis of motion (rotary or sliding) of joint i+1.
- D4. Establish the origin of the ith coordinate system. Locate the origin of the ith coordinate at the intersection of the Zi & Zi-1 or at the intersection of common normal between the Zi & Zi-1 axes and the Zi axis.
- D5. Establish Xi axis. Establish or along the common normal between the Zi-1 & Zi axes when they are parallel.
- D6. Establish Yi axis. Assign to complete the right-handed coordinate system.

Denavit-Hartenberg Convention

- D7. Establish the hand coordinate system
- D8. Find the link and joint parameters : d,a,a,q

D-H transformation for adjacent coordinate frames:

Example (3.3):

Link Frame Assignments

Example:SCARA Robot

- The location of the sliding axis Z2is arbitrary, since it is a free vector. For simplicity, we make it coincident with Z3 . thus a2and d2are arbitrarily set.
- The placement of O3and X3along Z3is arbitrary, since Z2and Z3are coincident. Once we choose O3, however, then the joint displacement d3is defined.
- We have also placed the end link frame in a convenient manner, with the Z4axis coincident with the Z3axis and the origin O4displaced down into the gripper by d4.

Example: Puma 560

Different Configuration

Link Coordinate Parameters

PUMA 560 robot arm link coordinate parameters

The Tool Transform

- A robot will be frequently picking up objects or tools.
- Standard practice is to to add an extra homogeneous transformation that relates the frame of the object or tool to a fixed frame in the end-effector.

Kinematic Calibration

How one knows the DH parameters?

- Certainly when robots are built, there are design specifications. Yet due to manufacturing tolerances, these nominal parameters will not be exact.
- The process of kinematic calibrationdetermines these nominal parameters experimentally. Kinematic calibration is typically accomplished with an external metrology system, although alternatives that do not require a metrology system exist.

You are have a three link arm that starts out aligned in the x-axis. Each link has lengths l1, l2, l3, respectively. You tell the first one to move by 1, and so on as the diagram suggests. Find the Homogeneous matrix to get the position of the yellow dot in the X0Y0 frame.

Y3

3

2

3

X3

Y2

2

H = Rz(1) * Tx1(l1) * Rz(2) * Tx2(l2) * Rz(3)

i.e. Rotating by 1will put you in the X1Y1frame.

Translate in the along the X1 axis by l1.

Rotating by 2will put you in the X2Y2frame.

and so on until you are in the X3Y3frame.

The position of the yellow dot relative to the X3Y3frameis

(l1, 0). Multiplying H by that position vector will give you the

coordinates of the yellow point relative the the X0Y0frame.

X2

Y0

1

X1

1

Y1

X0

Visual Approach - “A way to visualize the link parameter a(i-1) is to imagine an expanding cylinder whose axis is the Z(i-1) axis - when the cylinder just touches the joint axis i the radius of the cylinder is equal to a(i-1).”(Manipulator Kinematics)

DH convention: Assign Z axes

- Use actuation as a guide
- Prismatic – joint slides along zi
- Revolute – joint rotates around zi
- Establish base frame {0}:
- Nearly arbitrary
- Start at base and assign frames 1,…,N
- Pick x-axis and origin
- y-axis chosen to form a right hand system

DH convention: Assign Z axes

- Use actuation as a guide
- Prismatic – joint slides along zi
- Revolute – joint rotates around zi
- Establish base frame {0}:
- Nearly arbitrary
- Start at base and assign frames 1,…,N
- Pick x-axis and origin
- y-axis chosen to form a right hand system

Robot Base

- Often base is “given” or some fixed point on the work-table is used.
- z0 is along joint axis 1
- Original:
- any point on z0 for origin
- Modified DH:
- {0} is defined to be completely co-incident with the reference system {1}, when the variable joint parameter, d1 or q1 , is zero.

DH convention: Assign X axes

- Start at base and assign frames 1,…,N
- Pick x-axis and origin
- y-axis chosen to form a right hand system
- Consider 3 cases for zi-1 and zi:
- Not-coplanar
- Parallel
- Intersect

DH convention: x axis

- zi-1 and zi are not-coplanar
- Common normal to axes is the “link” axis
- Intersection with zi is origin

Usually, xi points from frame i-1 to i

zi-1

Xi

zi

DH convention: x axis

- zi and zi-1 are parallel
- Infinitely many common normals
- Pick one to be the “link” axis
- Choose normal that passes through origin of frame {i-1} pointing toward zi
- Origin is intersection of xi with zi

Xi

zi-1

zi

DH convention: x axis

zi

If joint axes zi-1 and zi intersect, xi is normal to the plane containing the axes

xi = (zi-1 zi )

zi-1

link i

Xi

DH convention: Origin non-coplanar Z

Origin of frame {i} is placed at intersection of joint axis and link axis

zi

xi

DH convention: Origin parallel Z

- zi and zi-1 are parallel
- Origin is intersection of xi with zi

zi-1

zi

xi

DH convention: x axis - parallel Z

- zi and zi-1 are parallel
- Origin is intersection of xi with zi
- Yi is chosen to make a right hand frame

yi

zi-1

zi

xi

DH convention: origin

If joint axes intersect, the origin of frame {i} is usually placed at intersection of the joint axes

zi

zi-1

link i

xi

Z2

Z1

Z0

X2

d2

X0

X1

Y0

Y1

a0

a1

3 Revolute Joints

Denavit-Hartenberg Link Parameter Table

Notice that the table has two uses:

1) To describe the robot with its variables and parameters.

2) To describe some state of the robot by having a numerical values for the variables.

DH Example: “academic manipulator”

3 revolute joints

Shown in home position

joint 1

R

Link 2

Link 3

Link 1

joint 2

joint 3

L1

L2

DH Example: “academic manipulator”

Zi is axis of actuation for joint i+1

Z0

Z0 and Z1 are not co-planar

Z1 and Z2 are parallel

1

3

2

Z1

Z2

DH Example: “academic manipulator”

Z0 and Z1 are not co-planar:

x0 is the common normal

Z0

1

x1

x2

x3

x0

3

2

Z3

Z1

Z2

DH Example: “academic manipulator”

Z0 and Z1 are not co-planar:

x0 is the common normal

Z0

1

x1

x2

x3

x0

3

2

Z3

Z1

Z2

Z1 and Z2 are parallel :

x1 is selected as the common normal that lies along the center of the link

DH Example: “academic manipulator”

Z0 and Z1 are not co-planar:

x0 is the common normal

Z0

1

x1

x2

x3

x0

3

2

Z3

Z1

Z2

Z2 and Z3 are parallel :

x2 is selected as the common normal that lies along the center of the link

DH Example: “academic manipulator”

Shown with joints in non-zero positions

Z0

x3

z3

3

2

x2

x1

Z2

1

x0

Z1

Observe that frame i moves with link i

DH Example: “academic manipulator”

Link lengths given

1 = 90o(rotate by 90o around x0 to align Z0 and Z1)

R

Z0

L2

L1

x1

x2

x3

1

x0

Z3

Z1

Z2

x3

z3

3

2

x2

x1

z2

1

x0

z1

DH Example: “academic manipulator”x1 axis expressed wrt {0}

y1 axis expressed wrt {0}

z1 axis expressed wrt {0}

Origin of {1} w.r.t. {0}

x3

z3

3

2

x2

x1

z2

1

x0

z1

DH Example: “academic manipulator”x2 axis expressed wrt {1}

y2 axis expressed wrt {1}

z2 axis expressed wrt {1}

Origin of {2} w.r.t. {1}

x3

z3

3

2

x2

x1

z2

1

x0

z1

DH Example: “academic manipulator”x3 axis expressed wrt {2}

y3 axis expressed wrt {2}

z3 axis expressed wrt {2}

Origin of {3} w.r.t. {2}

DH Example: “academic manipulator”– alternate end-effector frame

Zi is axis of actuation for joint i+1

Z0

Z0 and Z1 are not co-planar

Z1 and Z2 are parallel

1

Pick this z3

3

2

Z1

Z2

DH Example: “academic manipulator”– alternate end-effector frame

Z0

y2

1

x1

x2

x0

1

Z3

3

2

Z1

Z2

Would need to rotate about y2 here!

DH Example: “academic manipulator”– alternate end-effector frame

Z0

x’2

1

x1

x2

x0

1

Z3

3

2

Z1

Solution: Add “offset” to rotation about z2

(q3+90o )

DH Example: “academic manipulator”– alternate end-effector frame

Z0

x’2

x3

L2

1

x1

x2

x0

1

Z3

3

2

Z1

Z2

Now can rotate about x’ to align z2 and z3

DH Example: “academic manipulator”– alternate end-effector frame

Inverse Kinematics

Summary: Robot Forward Kinematics Analysis

- 1. Identify robot dimensions and understand its geometry.
- 2. Number Links and Joints.
- Base = Link 0
- 1st Joint = Joint 1
- 3. Identify Joint Axes, type and direction of positive motions.
- 4. Draw Common Normals (CNs) and find their intersections
- with joint axes. Define two points:
- AN = intersection of Axis N with CN to Axis N+1
- BN = intersection of Axis N with CN to Axis N-1
- 5. Assign Link Frames.
- Origin of Frame N is at AN
- ZN points along Axis N.
- XN points along CN to BN+1.
- YN completes a right handed coordinate system.
- Q:What if there is no CN because the axes intersect?
- A:Then choose XN to be normal to ZN and ZN+1.
- Q:What if there is no unique CN because ZN and ZN+1 are parallel
- A:Then choose one. Typically, you can simplify subsequent
- analysis by choosing a CN which intersects BN.

6. Derive the Denavit Hartenburg Parameters with respect to

- the link parameters for each link.
- = distance from AN to BN+1
- = distance from ZN to ZN+1 along XN
- = “link length”
- = angle between Axis N and Axis N+1 about CN
- = angle between ZN and ZN+1 about XN
- = “link twist”

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