CHAPTER 3 ROBOT CLASSIFICATION DAE 32503 – ROBOTICS & AUTOMATION SYSTEM By: Nor Faezah Adan

ROBOT GEOMETRY 1 KINEMATICS & PLANNING 2 DYNAMICS &CONTROL 3

REFERENCES Craig, John J. (2005). Introduction to robotics : Mechanics and control. Pearson. Shelf No.: TJ211 .C72 2005

Robot Geometry: Degrees of Freedom (DOF) For each degree of freedom, a joint is required. 1 joint = 1 DOF The more degrees of freedom, the greater the complexity of motions encountered. For applications that require more flexibility, additional degrees of freedom are used in the wrist of the robot. Three degrees of freedom located in the wrist give the end effector all the flexibility. 6 DOF

Robot Geometry: Degrees of Freedom (DOF) How many DOF?

Robot Geometry: Degrees of Freedom (DOF) How many DOF?

Robot Geometry: Degrees of Freedom (DOF) The 3 DOF located in the arm of a robotic system: The rotational traverse The rotational traverse is the movement of the arm assembly about a rotary axis, such as the left-and-right swivel of the robot’s arm on a base. The radial traverse The radial traverse is the extension and retraction of the arm or the in-and-out motion relative to the base.

Robot Geometry: Degrees of Freedom (DOF) The vertical traverse The vertical traverse provides the up-and-down motion of the arm of the robotic system.

Robot Geometry: Degrees of Freedom (DOF)

Robot Geometry: Degrees of Freedom (DOF) The 3 DOF located in the wrist of a robotic system: Pitch Bend or up and down movement. Yaw Right and left movement. Roll Swivel or rotation of the wrist/hand.

Robot Geometry: Degrees of Freedom (DOF)

Robot Geometry: Robot configurations & Work envelope In general, the fundamental mechanical configurations of robot manipulators are categorized as Cartesian, Cylindrical, Spherical and Articulated / Jointed-arm. Cartesian is divided into traverse & gantry types. Articulated is divided into horizontal & vertical types.

Robot Geometry: Robot configurations & Work envelope Work envelope / workspace The extreme position of the robot axes describe a boundary for the region in which the robot operates. This boundary encloses the work envelope. The size of a work envelope determines the limits of reach.

Cartesian / Rectangular robot Robot Geometry: Robot configurations & Work envelope Cartesian / Rectangular robot It has 3 prismatic joints, whose axes are coincident with a cartesian coordinate system. Most cartesian robots come as Gantries, distinguished by a frame structure supporting the linear axes. Gantry robots are widely used for: Special machining tasks such as water jet or laser cutting where robot motion cover large surfaces. Palletizing Warehousing

Robot Geometry: Robot configurations & Work envelope Gantry Traverse

Work envelope: Cartesian /rectangular robot Robot Geometry: Robot configurations & Work envelope Work envelope: Cartesian /rectangular robot Shaped as a cube or a rectangle.

Robot Geometry: Robot configurations & Work envelope

The manipulator has 2 linear motions and 1 rotary motion. Robot Geometry: Robot configurations & Work envelope Cylindrical robot The manipulator has 2 linear motions and 1 rotary motion. Robot’s manipulator has 1 rotational degree of freedom and 2 translational (linear) degrees of freedom. A cylindrical-coordinated robot generally results in a larger work envelope than cartesian-coordinated robot. This robot is ideally suitable for pick and place operation. Typical applications are assembly, conveyor pallet transfer, palletizing etc.

Robot Geometry: Robot configurations & Work envelope Cylindrical

Work envelope: Cylindrical robot Robot Geometry: Robot configurations & Work envelope Work envelope: Cylindrical robot It can move it’s gripper within a volume described by a cylinder.

Robot Geometry: Robot configurations

Spherical / Polar robot Robot Geometry: Robot configurations & Work envelope Spherical / Polar robot The manipulator has 1 linear motion and 2 rotary motions. The first motion corresponds to base rotation. The second motion corresponds to an elbow rotation. The third motion corresponds to a radial/in-out/ translation. A spherical robot generally results in a larger work envelope than cylindrical and cartesian robot. This robot is ideally suitable for applications where a small amount of vertical movement is adequate such as loading & unloading a punch press.

Robot Geometry: Robot configurations Spherical

Robot Geometry: Robot configurations & Work envelope Work envelope: Spherical / Polar robot The envelope is shaped like a section of a sphere with upper and lower limits imposed by the angular rotations of the arm.

Robot Geometry: Robot configurations & Work envelope

Articulated / Jointed-arm robot - Vertical Robot Geometry: Robot configurations & Work envelope Articulated / Jointed-arm robot - Vertical The manipulator has 3 rotary motions to reach any point in space. The design is similar to human arm. The first rotation is about the base, the second rotation is about the shoulder in a horizontal axis and the final motion is rotation about the elbow. It can move at high speeds and has a greater variety of angles to approach a given point and thus very useful for painting and welding applications.

Robot Geometry: Robot configurations & Work envelope

Robot Geometry: Robot configurations & Work envelope Work envelope: Vertical articulated/jointed-arm robot The envelope is circular when viewed from the top of the robot. When looked from the side, the envelope has a circular outer surface with an inner scalloped surface.

Robot Geometry: Robot configurations & Work envelope

Articulated / Jointed-arm robot - Horizontal Robot Geometry: Robot configurations & Work envelope Articulated / Jointed-arm robot - Horizontal The manipulator has 2 rotary motions and 1 linear (vertical) motion to reach any point in space. Also called SCARA (Selective Compliance Assembly Robot Arm). This robot has 2 horizontally jointed-arm segments fixed to a rigid vertical member (base) and one vertical linear motion axis. It is extremely useful in assembly operations where insertions of objects into holes are required.

Robot Geometry: Robot configurations & Work envelope

Work envelope: Horizontal articulated/jointed-arm robot Robot Geometry: Robot configurations & Work envelope Work envelope: Horizontal articulated/jointed-arm robot

Robot Geometry: Robot configurations & Work envelope

Configuration Advantages Disadvantages Cartesian coordinates x, y, z (base travel, reach, and height) Three linear axes Easy to visualize Rigid structure Easy to program off-line Linear axes make for easy mechanical stops Can only reach in front of itself Requires large floor space for size of work envelope Axes hard to seal Cylindrical coordinates θ, y, z – (base rotation, reach, and height) Two linear axes, one rotating axis Can reach all around itself Reach and height axes rigid axis Rotation axis easy to seal Cannot reach above itself Base rotation axis is less rigid than a linear Linear axis is hard to seal Won’t reach around obstacles Horizontal motion is circular Spherical coordinates (vertical) θ, y, β (base rotation, elevation angle, reach angle) One linear axis, two rotating axes Long horizontal reach Can’t reach around obstacles Generally has short vertical reach Articulated (or jointed-arm) coordinates (vertical) θ, β, α (base rotation, elevation angle, reach angle) Three rotating axes Can reach above or below obstacles Largest work area for least floor space Two or four ways to reach a point Most complex manipulator SCARA coordinates (horizontal) θ, Φ, z (base rotation, reach angle, height) Height axis is rigid Large work area for floor space Can reach around obstacles Two ways to reach a point Difficult to program off-line Highly complex arm

Robot Geometry: Work envelope Summary of work envelope

Robot Geometry: Work envelope End of Lecture 6

Kinematics & Planning: Transformations Kinematics is the science of motion that treats motion without regard to the forces which causes it. Coordinate systems - Relative frames Consider the problem of a robot holding a part for insertion into several CNC machines for various operations (drilling/grinding). The robot first grasp the part in a specified way and inserts it into the first machine. After the first machining operation, the robot grasps the part in a different way and inserts it into the second machine. The problem is, how to exactly specify exactly those 2 gripping positions?

Kinematics & Planning: Transformations Example: The vector defining C is given in {W} frame. We may need to transform this into the robot base co-ordinate frame {B} and/or into the end effector frame {E}.

Kinematics & Planning: Transformations Point P is located in coordinate frame {A}. The position vector representing P:

Kinematics & Planning: Transformations Mappings: Changing descriptions from frame to frame Describe frames Pure translation Transform point C vector: Pure translation from frame {W} to {B}. **Frame {B} and {W} have the same orientation.

Kinematics & Planning: Transformations

Kinematics & Planning: Transformations Pure rotation Transform point C vector: Pure rotation of to . **Frame {B} and {W} have the same origin position.

Kinematics & Planning: Transformations Rotation around z-axis:

Kinematics & Planning: Transformations These equations can be expressed in matrix form:

Kinematics & Planning: Transformations

Kinematics & Planning: Transformations Summary Rotation matrices:

Kinematics & Planning: Transformations Exercise Pure translation Answer

Kinematics & Planning: Transformations Exercise Pure rotation at θ=30o Answer

Kinematics & Planning: Transformations Combined Translation & Rotation Frame {B} is not coincident with frame {A} but has a general vector offset which is the vector that locates {B}’s origin, . Also, {B} is rotated with respect to {A}, as described by . Given , we want to compute .

Kinematics & Planning: Transformations

Kinematics & Planning: Transformations Homogenous Transformation Matrix or 4x4 Transformation Matrix

Kinematics & Planning: Transformations Operators: Translations, Rotations & Transformations  Translate points, rotate vector or both. Translational operators

Kinematics & Planning: Transformations Rotational operators

Kinematics & Planning: Transformations Transformation operators Operator T rotates and translates a vector to compute a new vector .

Kinematics & Planning: Transformations Combined transformations relative to fixed reference frame A point P(7,3,2)T is attached to a frame {W} and is subjected to the transformations described below, all relative to a reference frame {B}. Find the coordinates of the point relative to the fixed reference frame. Rotation of 90 degrees about z-axis. Followed by rotation of 90 degrees about the y-axis. Followed by a translation of (4,-3,7) T.

Kinematics & Planning: Transformations

Kinematics & Planning: Transformations Combined transformations relative to fixed reference frame A point P(7,3,2)T is attached to a frame {W} and is subjected to the transformations described below, all relative to a reference frame {B}. Find the coordinates of the point relative to the fixed reference frame. Rotation of 90 degrees about z-axis. Followed by a translation of (4,-3,7) T. Followed by rotation of 90 degrees about the y-axis.

Kinematics & Planning: Transformations

Kinematics & Planning: Transformations Combined transformations relative to rotating frame A point P(7,3,2)T is attached to a frame {W} and is subjected to the transformations described below but all relative to the current moving frame {W}. Find the coordinate of the point relative to the reference frame {B} after transformations are completed. Rotation of 90 degrees about z-axis. Followed by rotation of 90 degrees about the y-axis. Followed by a translation of (4,-3,7) T.

Kinematics & Planning: Transformations

Kinematics & Planning: Transformations Combined transformations relative to rotating frame A point P(7,3,2)T is attached to a frame {W} and is subjected to the transformations described below but all relative to the current moving frame {W}. Find the coordinate of the point relative to the reference frame {B} after transformations are completed. Rotation of 90 degrees about z-axis. Followed by a translation of (4,-3,7) T. Followed by rotation of 90 degrees about the y-axis.

Kinematics & Planning: Transformations

Kinematics & Planning: Transformations Compound transformations

Kinematics & Planning: Transformations Frame {C} is relative to frame {B}. Frame {B} is relative to frame {A}.

Kinematics & Planning: Transformations Inverting a transformation matrix Consider frame {B} that is known relative to frame {A}, i.e we know the value of ATB , sometimes, we will wish to invert this transform in order to get description of {A} relative to {B}. End of Lecture 7

Kinematics & Planning: Forward Kinematics Forward kinematics is used to determine the location of an end effector with respect to a reference coordinate frame. Denavit Hartenberg convention (D-H) Assumptions: Robots may be made of a succession of joint and links. Joints may be either prismatic or revolute. Joints may be in any order or sequence and may be in any plane. Links may also be of any length including zero, maybe twisted or bent and may be in any plane.

Kinematics & Planning: Forward Kinematics Assume we are at local reference frame Xn - Zn, we will do the following four standard steps to get to the next local reference frame Xn+1 - Zn+1 . Rotate about Zn an angle of θn+1 (this will make Xn and Xn+1 parallel to each other). Translate along Zn a distance dn+1 to make Xn and Xn+1 collinear. Translate along Xn+1 a distance of an+1 to bring the origins of both frames together. Rotate Zn axis about Xn+1 axis an angle of αn+1 to align Zn axis with axis Zn+1 .

Kinematics & Planning: Forward Kinematics Positive rotation is clockwise

Kinematics & Planning: Forward Kinematics

Kinematics & Planning: Forward Kinematics Example: 3 DOF robot 

Kinematics & Planning: Forward Kinematics

Kinematics & Planning: Forward Kinematics

Kinematics & Planning: Inverse Kinematics Inverse kinematics Inverse kinematics is the calculation of the joint variables from the end effector position and orientation. In forward kinematic, given the overall transformation matrix is In inverse kinematics, we will pre-multiply the RTH matrix with individual An-1 in order to calculate the angles, θn.

Kinematics & Planning: Inverse Kinematics Example: 3 DOF robot. End of Lecture 8

Kinematics & Planning: Dynamics & Control Differential motion Suppose you have a robot welding two pieces together. For best results, you want the robot to move at a constant speed. This means that the differential motions of the hand frame must be defined to represent a constant speed in a particular direction. This relates to the differential motion of the frame. However, the motion is caused by the robot. Thus, we have to calculate the speeds of each and every joint at any instant, such that the total motion caused by the robot will equal to the desired speed of the frame.

Kinematics & Planning: Dynamics & Control Differential motions of a frame Differential translations A differential translation is a translation of a frame at differential values. They can be represented by TRANS (dx,dy,dz). This means that the frame has moved a differential amount along the three axes.

Kinematics & Planning: Dynamics & Control Differential rotations A small rotation of the frame. Since the rotations are small: Sin δx = δx (in radians) Cos δx = 1

Kinematics & Planning: Transformations Differential rotation matrices:

Kinematics & Planning: Transformations Differential rotation operator =

Kinematics & Planning: Dynamics & Control Differential transformations The differential transformation of a frame is a combination of differential translations & rotations. Differential operator

Kinematics & Planning: Dynamics & Control Interpretation of the differential change The differential operator represents the changes in a frame a result of differential motions. As a result of the changes, the new location and orientation of the frame is given as:

Kinematics & Planning: Dynamics & Control The differential motions of a robot and its hand frame Previous slides mentioned about changes made to a frame as a result of differential motions. This only relates to the frame changes only, not how they were accomplished. In this section we will relate the changes to the mechanism that accomplishes the differential motions i.e we will learn how the robot’s movements are translated into the frame changes.

Kinematics & Planning: Dynamics & Control Consider the frame T discussed previously as the hand frame of the welding robot. We will need to find out how the differential motions of the joints of the robot would relate to the differential motions of the hand frame, dT. The relationship between the joint movements and the hand movement can be linked using the robot JACOBIAN. ROBOT JACOBIAN

Kinematics & Planning: Dynamics & Control How to relate the JACOBIAN and the differential operator Example: A frame of a robot with 5 DOF, its numerical Jacobian for this instant, and a set of differential motions are given. Find the new location of the hand after the differential motion.

Kinematics & Planning: Dynamics & Control Solution: Use [D]=[J][Dθ] to find dx,dy,dz,δx and δy. Substitute the values into the differential operator matrix. Use [dT]=[∆][T] to get the differential motion of the frame. Use Tnew = Toriginal + dT to get the new location of the frame after the differential motion. Final answer  End of Lecture 9

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