Dr. Ibrahim Al-Naimi - Philadelphia University · 2018-12-02 · •A manipulator is composed of...

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Robotics and automation

Dr. Ibrahim Al-Naimi

Chapter four

Forward Kinematics

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Introduction to Forward Kinematics

• Kinematics studies the motion of bodies without consideration of the forces or moments that cause the motion.

• A manipulator is composed of serial links which are affixed to each other revolute or prismatic joints from the base frame through the end-effector.

• Calculating the position and orientation of the end-effector in terms of the joint variables is called forward kinematics.


• The forward kinematics comprises changing the representation of manipulator position from a joint space description into a Cartesian space (task space) description.

• The joint variables are the angles between the links in the case of revolute or rotational joints, and the link extension in the case of prismatic or sliding joints.

• In order to have forward kinematics for a robot mechanism in a systematic manner, one should use a suitable kinematics model.

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• Denavit-Hartenberg (DH) method is the most common method for describing the robot kinematics.

• Denavit and Hartenberg (1955) showed that a general transformation between two joints requires four parameters. These parameters known as the Denavit-Hartenberg (DH) parameters have become the standard for describing robot kinematics.


• These parameters ai-1, αi−1 , di and θi are the link length, link twist, link offset and joint angle, respectively.

• A coordinate frame is attached to each joint to determine DH parameters.

• These parameters are used to formulate the transformation matrices. Thus, the final transformation matrix that descries the position and orientation of end-effector frame with respect to the base frame is determined.

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Overall, the following steps are followed to determine the forward kinematics of a robotic manipulator. – Attach an inertial frame to the robot base.

– Attach frames to links ( or joints), including the end-effector.

– Determine the DH parameters for each link.

– Determine the homogenous transformation between each frame using the DH parameter.

– Apply the set of transforms sequentially to obtain a final overall transform (i.e. Between the end-effector and the robot base).

Notations and Description of Kinematic Chains

• A robot manipulator is composed of a set of links connected together by joints;

• Joints can be either:

– Revolute joint (a rotation by an angle about fixed axis)

– prismatic joint (a displacement along a single axis) More complicated joints (of 2 or 3 degrees of freedom) can always be thought of as a succession of single degree-of-freedom joints with links of length zero in between.

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• A robot manipulator with n joints will have (n + 1) links. Each joint connects two links.

• We number joints from 1 to n, and links from 0 to n. So that joint i connects links (i − 1) and i.

• The location of joint i is fixed with respect to the link (i−1).

• When joint i is actuated, the link i moves. Hence the link (i-1) is fixed. In addition, link 0 is fixed.


• With the ith joint, we associate joint variable qi:

• Joint space (joint vector) is a set of all joint variables in the manipulator.

• For each link we attach rigidly the coordinate frame (oi xi yi zi) for the link i. (Using right hand rule).

• When joint i is actuated, the link i and its frame experience a motion.

• The frame (o0 x0 y0 z0) attached to the base is referred to as inertial frame (base frame)

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• Suppose Ai is the homogeneous transformation that gives the position and orientation of frame (oi xi yi zi) with respect to frame (oi−1 xi−1 yi−1 zi−1).

• The matrix Ai is changing as robot configuration changes.

• Due to the assumptions Ai = Ai(qi), i.e. it is the function of a scalar variable.

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• Homogeneous transformation that expresses the position and orientation of (oj xj yj zj) with respect to (oi xi yi zi) is called a transformation matrix.

• The position and orientation of the end-effector with respect to the inertia frame are

Denavit-Hartenberg (DH) Representation

• While it is possible to carry out all of the analysis in this chapter using an arbitrary frame attached to each link, it is helpful to be systematic in the choice of these frames.

• A commonly used convention for selecting frames of reference in robotic applications is the Denavit-Hartenberg, or D-H convention.

• In this convention, each homogeneous transformation Ai is represented as a product of four basic transformations.

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• where the four quantities θi, ai, di, αi are parameters associated with link i and joint i.

• The four parameters ai, αi, di, and θi are generally given the names link length, link twist, link offset, and joint angle, respectively.


• These names derive from specific aspects of the geometric relationship between two coordinate frames.

• Since the matrix Ai is a function of a single variable, it turns out that three of the above four quantities are constant for a given link, while the fourth parameter, θi for a revolute joint or di for a prismatic joint, is the joint variable.

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• As discussed in the previous chapter, an arbitrary homogeneous transformation matrix can be characterized by 12 numbers, or by 6 numbers (i.e. three numbers to specify the fourth column of the matrix and three Euler angles to specify the upper left 3 × 3 rotation matrix).

• In the D-H representation, in contrast, there are only four parameters used to describe the position and orientation of frame. How is this possible?


• The answer is that, while frame i is required to be rigidly attached to link i, we have considerable freedom in choosing the origin and the coordinate axes of the frame.

• For example, it is not necessary that the origin, oi, of frame i be placed at the physical end of link i. In fact, it is not even necessary that frame i be placed within the physical link; frame i could lie in free space — so long as frame i is rigidly attached to link i.

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• By a clever choice of the origin and the coordinate axes, it is possible to cut down the number of parameters needed from six to four.

• DH constrains:

– (DH1) The axis xi is perpendicular to the axis zi-1

– (DH2) The axis xi intersects the axis zi-1

• Only joints with a single degree of freedom are Considered in DH conventions. Joints of a higher order can be modelled as a combination of single degree of freedom joints.


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• The links and joints are numbered starting from the immobile base of the robot, referred to as link 0, continuing along the serial chain in a logical fashion.

• The first joint, connecting the immobile base to the first moving link is labeled joint 1, while the first movable link is link 1. Numbering continues in a logical fashion.

• The geometrical configuration of the manipulator can be described by 4 parameters: ai, αi, di, and θi .


Link length, ai-1

• For any two axes in 3-space, there exists a well-defined measure of distance between them. This distance is measured along a line that is mutually perpendicular to both axes. This mutual perpendicular always exists; it is unique except when both axes are parallel, in which case there are many mutual perpendiculars of equal length.

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Link length, ai-1

• Consider the shortest distance between the axis of joint i-1 and joint i in the following figure. This distance is realized along the vector mutually perpendicular to each axis and connecting the two axes.

• The length of this vector is the link length ai-1.

• Note that link length need not be measured along a line contained in the physical structure of the link. Although only the scalar link length is needed in the mathematical formulation of joint transformations, the vector direction between joint axes is also important in understanding the geometry of a robotic manipulator.


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Link twist, αi-1

• Consider the plane orthogonal to the link length ai-

1. Both axes vectors of joint i-1 and i lie in this plane. Project the axes vectors of joints i and i-1 onto this plane.

• The link twist is the angle measured from joint axis i-1 to joint axis i in the right-hand sense around the link length ai-1.

• Direction of ai-1 taken as from axis i-1 to i. This is to say that αi-1 will be positive when the link twist (by the right-hand rule) is in the positive direction of ai-

1.


Link twist, αi-1

• Another way to visualise the link parameters (i.e. ai-1 and αi-1 ) is to imagine an expanding cylinder whose axis is the joint i-1 axis. when it just touches joint axis i, the radius of the cylinder is equal to ai-1

• The concept of an expanding cylinder (video).

• ai-1 and αi-1 can be used to define the relationship between any two lines (in this case axes) in space.

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Example: The figure shows the mechanical drawings of a robot link. If

this link is used in a robot, with bearing "A" used for the lower-numbered joint, give the length and twist of this link. Assume that holes are centered in each bearing.


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Link Offset ,di

• On the axis of joint i, consider the two points at which the link lengths ai-1 and ai are attached. The distance between these points is the link offset, measured positive from the ai-1 to ai connection points.

Joint Angle, θi

• Consider a plane orthogonal to the joint axis i. By construction, both link length vectors ai-1 and ai lie in this plane. The joint angle is calculated as the clockwise angle that the link length ai-1 must be rotated to be collinear with link length ai. This corresponds to the right-hand rule of a rotation of link length ai-1 about the directed joint axis.


• The following figure shows the interconnection of link i-1 and link i. Recall that ai-1 is the mutual perpendicular between the two axes of link i-1. Likewise, ai is the mutual perpendicular defined for link i. The first parameter of interconnection is the link offset, di which is the signed distance measured along the axis of joint i from the point where ai-1 intersects the axis to the point where ai intersects the axis. The link offset di is variable if joint i is prismatic. The second parameter of interconnection is the angle made between an extension of ai-1 and ai measured about the axis of joint i. The lines with the double hash marks are parallel. This parameter is named θi and is variable for a revolute joint.

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Denavit-Hartenberg (DH) Representation Example: Two links, as described in the figure, are connected as

links 1 and 2 of a robot. Joint 2 is composed of a "B" bearing of link 1 and an "A" bearing of link 2, arranged so that the flat surfaces of the "A" and "B" bearings lie flush against each other. What is d2?


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First and last links in the chain (DH convention):

• an = a0 = 0 and αn = α0 = 0.

• If joint 1 is revolute, the zero position for θ1 may be chosen arbitrarily and d1 = 0.0

• If joint 1 is prismatic, the zero position for d1 may be chosen arbitrarily and θ 1 = 0.0

• The same statements apply to joint n.

Convention for affixing Frames to links:

Link-frame attachment procedure

1. Identify the joint axes and imagine (or draw) infinite lines along them. For steps 2 through 5 below, consider two of these neighbouring lines (at axes i and i + 1).

2. Identify the common perpendicular between them, or point of intersection. At the point of intersection, or at the point where the common perpendicular meets the ith axis, assign the link-frame origin.

3. Assign the Zi axis pointing along the ith joint axis.

4. Assign the Xi axis pointing along the common perpendicular, or, if the axes intersect, assign Xi to be normal to the plane containing the two axes.

5. Assign the Yi axis to complete a right-hand coordinate system.

6. Assign {0} frame to match {1} frame when the first joint variable is zero. For {N}, choose an origin location and XN direction freely, but generally so as to cause as many linkage parameters as possible to become zero.

7. Create a table of link parameters ai, di, αi, θi.

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Convention for affixing Frames to links:

The DH link parameters in terms of the link frames


Notes:

•We usually choose ai > 0, because it corresponds to a distance; however, αi, di, and θi are signed quantities.

•The convention outlined above does not result in a unique attachment of frames to links. For example, when we first align the Zi axis with joint axis i, there are two choices of direction in which to point Zi.

•Frame {0} - The frame attached to the base of the robot or link 0 called frame {0} This frame does not move and, for the problem of arm kinematics, it can be considered as the reference frame.

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Notes: • Frame {0} is arbitrary, so it always simplifies matters to choose

Z0 along axis 1 and to locate frame {0} so that it coincides with frame {1} when joint variable 1 is zero. Using this convention, we will always have a0 = 0.0, α0 = 0.0. Additionally, this ensures that d1 = 0.0 if joint 1 is revolute, or θ1 = 0.0 if joint 1 is prismatic. • For joint n revolute, the direction of XN is chosen so that it

aligns with XN-1 when θn= 0.0, and the origin of frame {N} is chosen so that dn= 0.0. For joint n prismatic, the direction of XN is chosen so that θn= 0.0, and the origin of frame {N} is chosen at the intersection of XN-1 and joint axis n when dn=0.0. • For a revolute joint, link offset is fixed and joint angle is a

controlled variable. For a prismatic joint, joint angle is fixed and link offset is a controlled variable.


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Example: Assign link frames and give the DH parameters for the RRR manipulator shown in the figure.


Solution:

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Solution:


Solution:

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Solution:


Solution:

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Denavit-Hartenberg (DH) Representation Solution:


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Example (2): Assign link frames and give the DH parameters for the manipulator shown in the figure.


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Solution (a):

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Solution (b):


Solution (c):

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Solution (d):



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Manipulator Forward Kinematics

Derivation of link transformations

• We wish to construct the transform that defines frame {i} relative to the frame {i - 1}. In general, this transformation will be a function of the four link parameters. For any given robot, this transformation will be a function of only one variable, the other three parameters being fixed by mechanical design. By defining a frame for each link, we have broken the kinematics problem into a sub problems. In order to solve each of these sub problems, namely we will further break each sub problem into four sub subproblems. Each of these four transformations will be a function of one link parameter only and will be simple enough that we can write down its form by inspection. We begin by defining three intermediate frames for each link—{P}, {Q}, and {R}.



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• If we wish to write the transformation that transforms vectors defined in {i} to their description in {i —1} we may write:

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Example:

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Concatenating Link Transformation

• Define link frames.

• Define DH parameters of each link.

• Compute the individual link transformation matrix.

• Relates frame { N } to frame { 0 }

• The transformation will be a function of all n joint variables.

• If the robot’s joint position sensors are queried, the Cartesian position and orientation of the last link may be computed by .


Actuator, Joint, and Cartesian Spaces

• The position of all the links of a manipulator of n degrees of freedom can be specified with a set of n joint variables. This set of variables is often referred to as the n x 1 joint vector. The space of all such joint vectors is referred to as joint space.

• We have been concerned with computing the Cartesian space description from knowledge of the joint-space description. We use the term Cartesian space when position is measured along orthogonal axes and orientation is measured according to any of the conventions outlined in the previous Chapter. Sometimes, the terms task-oriented space and operational space are used for what we will call Cartesian space.

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• So far, we have implicitly assumed that each kinematic joint is actuated directly by some sort of actuator. However, in the case of many industrial robots, this is not so. For example, sometimes two actuators work together in a differential pair to move a single joint, or sometimes a linear actuator is used to rotate a revolute joint, through the use of a four-bar linkage. In these cases, it is helpful to consider the notion of actuator positions. The sensors that measure the position of the manipulator are often located at the actuators, so some computations must be performed to realize the joint vector as a function of a set of actuator values, or actuator vector.



• As is indicated in the following figure, there are three representations of a manipulator's position and orientation: descriptions in actuator space, in joint space, and in Cartesian space.

• The ways in which actuators might be connected to move a joint are quite varied. For each robot we design or seek to analyze, the correspondence between actuator positions and joint positions must be solved.

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Frames with standards names

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• Base frame {B}: is located at the base of the manipulator. It is merely another name for frame {0}. It is affixed to a nonmoving part of the robot, sometimes called link 0.

• Station frame {S}: is located in a task-relevant location. In the previous figure, it is at the corner of a table upon which the robot is to work. As far as the user of this robot system is concerned, {S} is the universe frame, and all actions of the robot are performed relative to it. It is sometimes called the task frame, the world frame, or the universe frame. The station frame is always specified with respect to the base frame, that is



• Wrist frame {W}: is affixed to the last link of the manipulator. It is another name for frame {N}, the link frame attached to the last link of the robot. Very often, {W} has its origin fixed at a point called the wrist of the manipulator, and {W} moves with the last link of the manipulator. It is defined relative to the base frame—that is, {W}

• Tool frame {T}: is affixed to the end of any tool the robot happens to be holding. When the hand is empty, {T} is usually located with its origin between the fingertips of the robot. The tool frame is always specified with respect to the wrist frame. In the figure, the tool frame is defined with its origin at the tip of a pin that the robot is holding.

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• Goal frame {G}: is a description of the location to which the robot is to move the tool. Specifically this means that, at the end of the motion, the tool frame should be brought to coincidence with the goal frame. {G} is always specified relative to the station frame. In the following figure, the goal is located at a hole into which we want the pin to be inserted.



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Where is the tool?

• One of the first capabilities a robot must have is to be able to calculate the position and orientation of the tool it is holding with respect to a convenient coordinate system. That is, we wish to calculate the value of the tool frame, {T}, relative to the station frame, {S}. Once has been computed via the kinematic equations, we can use Cartesian transforms to calculate {T} relative to {S}. Solving a simple transform equation leads to


Where is the tool?

• This equation implements what is called the WHERE function in some robot systems. It computes "where" the arm is. For the situation in the previous figure, the output of WHERE would be the position and orientation of the pin relative to the table top.

• The previous equation can be thought of as generalizing the kinematics. computes the kinematics due to the geometry of the linkages, along with a general transform (which might be considered a fixed link) at the base end and another at the end-effector . These extra transforms allow us to include tools with offsets and twists and to operate with respect to an arbitrary station frame.

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Forward kinematics of Industrial Manipulator (PUMA 560 )

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• The kinematics equations of PUMA 560 specify how to compute the position and orientation of frame {6} (tool) relative to frame {0} (base) of the robot. These are the basic equations for all kinematic analysis of this manipulator.

Dr. Ibrahim Al-Naimi - Philadelphia University · 2018-12-02 · •A manipulator is composed of...

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