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**Robot Modeling and the Forward Kinematic Solution**

ME 4135 Lecture Series 4

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**Looking Closely at the T0n Matrix**

At the end of our discussion of “Robot Mapping” we found that the T0n matrix related the end of the arm frame (n) to its base (0) – Thus it must contain information related to the several joints of the robot It is a 4x4 matrix populated by complex equations for both position and orientation (POSE)

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**Looking Closely at the T0n Matrix**

To get at the equation set, we will choose to add a series of coordinate frames to each of the joint positions The frame attached to the 1st joint is labeled 0 – the base frame! – while joint two has Frame 1, etc. The last Frame is the end or n Frame

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**Looking Closely at the T0n Matrix**

As we have seen earlier, we can define a HTM (T(i-1)i) to the transformation between any two SO3 based frames Thus we will find that the T0n is given by a product formed by:

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**Looking Closely at the T0n Matrix**

For simplicity, we will redefine each of the of these transforms (T(i-1)i) as Ai Then, for a typical 3 DOF robot Arm, T0n = A1*A2*A3 While for a full functioned 6 DOF robot (arm and wrist) would be: T0n = A1*A2*A3*A4*A5*A6 A1 to A3 ‘explain’ the arm joint effect while A4 to A6 explain the wrist joint effects

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**Looking At The Frame To Frame Arrangements – Building A Modeling Basis**

When we move from one frame to another, we will encounter: Two translations (in a controlled sense) Two rotations (also in a controlled sense) A rotation and translation WRT the Zi-1 These are called the Joint Parameters A rotation and translation WRT the Xi These are called the Link Parameters

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**A model of the Joint Parameters**

NOTE!!!

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**A model of the Link Parameters**

ai or

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**Talking Specifics – Joint Parameters**

i is an angle measured about the Zi-1 axis from Xi-1 to Xi and is a variable for a revolute joint – its fixed for a Prismatic Joint di is a distance measured from the origin of Frame(i-1) to the intersection of Zi-1 and Xi and is a variable for a prismatic joint – its fixed for a Revolute Joint

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**Talking Specifics – Link Parameters**

ai (or li) is the Link length and measures the distance from the intersection of Zi-1 to the origin of Framei measured along Xi i is the Twist angle which measures the angle from Zi-1 to Zi as measured about Xi Both of these parameters are fixed in value regardless of the joint type. A Further note: Fixed does not mean zero degrees or zero length just that they don’t change

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**Very Important to note:**

Two Design Axioms prevail in this modeling approach Axiom DH1: The Axis Xi must be designed to be perpendicular to Zi-1 Axiom DH2: The Axis Xi must be designed to intersect Zi-1 Thus, within reason we can design the structure of the coordinate frames to simplify the math (they are under our control!)

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**Returning to the 4 ‘Frame-Pair’ Operators:**

1st is which is an operation of pure rotation about Z or: 2nd is d which is a translation along Z or:

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**Returning to the 4 Frame Operators:**

3rd is a Translation Along X or: 4th is which is a pure Rotation about X or:

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The Overall Effect is:

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**Simplifying this Matrix Product:**

This matrix is the general transformation relating each and every of the frame pairs along a robot structure

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**So, Since We Can Control the Building of this Set Of Frames, What Are The Rules?**

We will employ a method called the Denavit-Hartenberg Method It is a Step-by-Step approach for modeling each of the frames from the initial (or 0) frame all the way to the n (or end) frame This modeling technique makes each joint axis (either rotation or extension) the Z-axis of the appropriate frame (Z0 to Zn-1). The Joint motion, thus, is taken W.R.T. the Zi-1 axis of the frame pair making up the specific transformation matrix under design

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**The D-H Modeling Rules:**

Locate & Label the Joint Axes: Z0 to Zn-1 Establish the Base Frame. Set Base Origin anywhere on the Z0 axis. Choose X0 and Y0 conveniently and to form a right hand frame Then Repeat Steps 3 to 5 for i = 1 through i = n-1: Locate the origin Oi where the common normal to Zi-1 and Zi intersects Zi. If Zi intersects Zi-1 locate Oi at this intersection. If Zi-1 and Zi are parallel, locate Oi at Joint i+1

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**The D-H Modeling Rules:**

Establish Xi along the common normal between Zi-1 and Zi through Oi, or in the direction Normal to the plane Zi-1 – Zi if these axes intersect Establish Yi to form a right hand system Set i = i+1, if i<n loop back to step 3 Establish the End-Effector (n) frame: OnXnYnZn. Assuming the n-th joint is revolute, set kn = a along the direction Zn-1. Establish the origin On conveniently along Zn, typically mounting point of gripper or tool. Set jn = o in the direction of gripper closure (opening) and set in = n such that n = oxa. Note if tool is not a simple gripper, set Xn and Yn conveniently to form a right hand frame.

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**The D-H Modeling Rules:**

Create a table of “Link” parameters: i as angle about Zi-1 between X’s di as distance along Zi-1 i as angle about Xi between Z’s ai as distance along Xi Form HTM matrices A1, A2, … An by substituting , d, and a into the general model Form T0n = A1*A2*…*An

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**Some Issues to remember:**

If you have parallel Z axes, the X axis of the second frame runs perpendicularly between them When working on a revolute joint, the model will be simpler if the two X directions are in alignment at “Kinematic Home” – ie. Our model’s starting point To achieve this kinematic home, rotate the model about moveable axes (Zi-1’s) to align X’s Kinematic Home is not particularly critical for prismatic joints An ideal model will have n+1 frames However, additional frames may be necessary – these are considered ‘Dummy’ frames since they won’t contain joint axes

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**Applying D-H to a General Case:**

Case 1: Like a part of a biological arm And Note: Case 2: Like a part of a R-P manipulator

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**General Case: Considering Link i**

Connects Frames: i-1 and I and includes Joint i This information allowed us to ‘Build’ The L.P. (link parameter) Table seen here Frames Link Var d a S C S C i -1 to i i R + 37 17.5 u 47.8 u 17.8 0.306 .952 S( + 37) C( + 37)

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Leads to an Ai Matrix:

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**Lindeke’s Folly – The “Plastic Handed” Robot**

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**Frame Skeleton for Prismatic Hand Robot**

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**Depends on Location of n-frame!**

LP Table: Frames Link Var d a S C S C 0 → 1 1 R 1 -90 -1 S1 C1 1 → 2 2 2 6 0 S2 C2 2 → 3 3 3 0.5 90 S3 C3 3 → 4 4 P d S4 C4 4 → n 5 5 S5 C5 Depends on Location of n-frame!

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Leading to 5 Ai Matrices

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**#5 is: Now, Lets Form the FKS: T0n = A1*A2*A3*A4*A5**

This value is called the Hand Span and depends on the Frame Skeleton we developed Now, Lets Form the FKS: T0n = A1*A2*A3*A4*A5 Best to use a software: MathCad, Mathematica, etc.

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Solving for FKS Here we have a special case – two of the Joints are a “planer arm” revolute model – i.e. parallel, consecutive revolute joints These are contained in the A2 and A3 Matrices These should be pre-multiplied using a trigonometric tool that recognizes the sum of angle cases ((Full)Simplify in Mathematica or MathCad Simplify for A2*A3 then Evaluate for A1*{A2*A3}*A4*A5) Basically then: T0n = A1*{A2A3}*A4*A5

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**Extracts from Math Cad Analyses:**

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Continuing

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Continuing

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**In Tabular Form: Nx: C23C1C5 - S1S5 Ny: C1S5 + C23C5S1 Nz: -S23C5**

Ox: -C23C1S5 - S1C5 Oy: C1C5 - C23S1S5 Oz: S23S5 Ax: S23C1 Ay: S1S23 Az: C23 Dx: S23C1 + 6C1C S1 + S23C1(d4+4.25) Dy: S23S1 + 6S1C C1 + S23S1(d4+4.25) Dz: C23 - 6S2 + C23(d4+4.25)

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**Finalizing the FKS – perform a physical verification**

Physical verification means to plug known angles into the variables and compute the Ai’s and FKS against the Frame Skeleton

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**Physical Verification in MathCad:**

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**Physical Verification of FKS:**

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**Another? 6dof Articulating Arm – (The Figure Contains Frame Skelton)**

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LP Table Frames Link Var d a S C S C 0 → 1 1 R 1 90 -1 S1 C1 1 → 2 2 2 a2 0 S2 C2 2 → 3 3 3 a3 S3 C3 3 → 4 4 4 a4 -90 S4 C4 4 → 5 5 5 S5 C5 5 → 6 6 6* d6 S6 C6 * With End Frame in Better Kinematic Home, here, as shown it could be (6 - 90), which is a problem!

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A Matrices

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A Matrices, cont.

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Leads To: FKS of: After Preprocessing A2-A3-A4, to collect angular effect, the FKS must be reordered as A1-A234-A5-A6 and Simplified

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**Solving for FKS Pre-process {A2*A3*A4} with Evaluate Simplify**

They are the planer arm issue as in the previous robot model Then Form: A1* {A2*A3*A4}*A5*A6 Evaluate for FKS!

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**Simplifies to (in the expected Tabular Form):**

nx = C1·(C5·C6·C234 - S6·S234) - S1·S5·C6 ny = C1·S5·C6 + S1·(C5·C6·C234 - S6·S234) nz = S6·C234 + C5·C6·S234 ox = S1·S5·S6 - C1·(C5·S6·C234 + C6·S234) oy = - C1·S5·S6 - S1·(C5·S6·C234 + C6·S234) oz = C6·C234 - C5·S6·S234 ax = C1·S5·C234 + S1·C5 ay = S1·S5·C234 - C1·C5 az = S5·S234 dx = C1·(C234·(d6·S5 + a4) + a3·C23 + a2·C2) + d6·S1·C5 dy = S1·(C234·(d6·S5 + a4) + a3·C23 + a2·C2) - d6·C1·C5 dz = S234·(d6·S5 + a4) + a3·S23 + a2·S2

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Verify the Model Again, substitute known’s (typically 0 or 0 units) for the variable kinematic variables Check each individual A matrix against your robot frame skeleton sketch for physical agreement Check the simplified FKS against the Frame skeleton for appropriateness

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After Substitution: nx = C1·(C5·C6·C234 - S6·S234) - S1·S5·C6 = 1(1-0) – 0 = 1 ny = C1·S5·C6 + S1·(C5·C6·C234 - S6·S234) = 0+ 0(1 – 0) = 0 nz = S6·C234 + C5·C6·S234 = = 0 ox = S1·S5·S6 - C1·(C5·S6·C234 + C6·S234) = 0 – 1(0 + 0) = 0 oy = - C1·S5·S6 - S1·(C5·S6·C234 + C6·S234) = -0 – 0(0 + 0) = 0 oz = C6·C234 - C5·S6·S234 = 1 – 0 = 1 ax = C1·S5·C234 + S1·C5 = = 0 ay = S1·S5·C234 - C1·C5 = 0 – 1 = -1 az = S5·S234 = 0 dx = C1·(C234·(d6·S5 + a4) + a3·C23 + a2·C2) + d6·S1·C5 = 1*(1(0 + a4) + a3 + a2) + 0 = a4 + a3 + a2 so here is ( = 3.2 m) dy = S1·(C234·(d6·S5 + a4) + a3·C23 + a2·C2) - d6·C1·C5 = 0(1(0 + a4) + a3 + a2) – d6 = -d6 so here is -.25 m dz = S234·(d6·S5 + a4) + a3·S23 + a2·S2 = 0(0 + a4) = 0

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**And Physical Verification w/ Math Cad:**

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