Pivot Wheel Drive Crab with a Twist

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on the z up axis All angle calculations will be carried out in radians All. angular values expressed here will be in radians unless otherwise noted. Normally in Crab Mode all wheels will be aligned with the direction of travel and. all drives will be equally powered or ideally all driven at the same speed which. is not necessarily the same thing All wheels would therefore be pivoted at. from straight ahead, The orientation of each pivot wheel from the chassis CP may also be defined. relative to the direction of travel i i 1 4,FIGURE I Basic Crab Chassis. Crab Modes, There are two logical ways to control Crab Mode There are also two logical. ways to twist in Crab Mode Each of the two twist modes fits intuitively also. logically and philosophically with its own Crab Mode We will describe the two. Crab Modes before describing the Twist Modes,Crab Mode 1 Anchored to the chassis. In this mode the direction of the joystick s deviation from neutral. corresponds directly to the chassis s direction of travel relative to the. chassis s orientation Pushing the joystick straight ahead moves the. chassis straight forward 0 Throwing the joystick directly right. moves the chassis towards the chassis s right side 3 2 Pulling the. joystick straight back moves the chassis backwards A good mode. if chassis orientation means something, It is not clear that the drive motors would ever reverse in Crab 1.
Crab Mode 2 Free form but I know where I am really. I am certain this is how Team 118 worked in 2007 except maybe the. knowing part, In Crab 2 y direction joystick movement runs the drive motors forward. or reverse x direction joystick movement runs the steering motors left. and right A neutral x position means that the robot continues on its. current heading regardless of that heading s orientation with the chassis. x direction joystick movement changes heading without regard to chassis. orientation, There is a presumption here that one wheel will be the master and the. other three slaves One wheel must set the heading The other three need. to point in the same direction, This could be very intuitive for a driver looking for movement without. having to pay and attention to chassis orientation Visually we are. directly attuned to movement Attention to orientation requires thought. slowing the process, This is as far as I care to judge Crab 1 2 without data They are different. though Fortunately through the magic of programming we can in principle. accommodate both,Twist Modes, Crab 1 locks joystick orientation with chassis drive direction Twist 1.
should be consistent This can be accomplished by incorporating a Snake. Mode turn into a Crab Mode 1 Twist Such a turn is shown schematically. in Figure 2, Through the turn the orientation of the chassis relative to the direction of. travel never changes Chassis orientation does change relative to the. field and does so in an overt controlled manner, Twist 1 is really a turn not a twist but it is a Snake turn performed in. FIGURE 2 Twist 1 Schematic, In Crab 2 steering is relative to the chassis s current heading in relation to. the field There is no fixed relationship between joystick position and. chassis heading In Crab 2 it would make sense for Twist to rotate the. chassis without if possible changing the heading relative to the field. This would necessarily change, So Twist 2 is a real twist and not a turn Twist 2 is shown schematically in. FIGURE 3 Twist 2 Schematic, Twist 2 introduces a new steering regime to 1640 dynamic steering.
Up to this point including Twist 1 steering can be considered static. That is once the human controls are set this determines a robot response. steering angle drive speed in this case which does not change until the. human controls change, Twist 2 is different In order to rotate the chassis while retaining a. constant bearing relative to the field steering and drive speed need to be. dynamic Steering angle and drive speed changes continuously for each. wheel as a function of i for given human interface values This will. provide a unique challenge for the team, Did I say 2 twist modes What was I thinking There s a caveat. If the robot is stationary a 3rd twist is logical This simply sets all of the. wheels tangent to the circle that they describe and turn the chassis like a. turret Ken Au has already programmed this as an independent mode. This is also the logical limit of a Snake Mode turn but in Snake the. inside pivots turn an extra See Figure 4, In the configuration shown it would be necessary to run two motors in. reverse to rotate the chassis, We will see that Twist 3 is logically incorporated in Twist 2 without a. special effort,FIGURE 4 Twist 3 Stationary Chassis Rotation.
The Math behind Twist 1,Twist 1 is a generalized approach to Snake Mode. Figure 5 defines the basic geometry behind Twist 1 The chassis is represented by. a circle of radius h about a chassis CP There is an angular bearing of travel. relative to the nominal chassis orientation of There are n n 4 in our case but. this is not important here wheels distributed around the circle s radius only one. is shown each at a known fixed orientation i relative to the physical chassis. but at a variable orientation relative to the direction of travel i so that. Care needs to be taken to check that i remains in the range 0 2 add or subtract. 2 to if necessary, When driving in Crab Mode without twist all wheels are oriented at 2. relative to the chassis s straight ahead orientation. As with the earlier Snake Mode analysis it is useful to imagine a reference. wheel on the centerline relative to the direction of travel This reference wheel is. located h distance ahead of the chassis CP The centerline reference wheel would. respond directly and proportionally to the joystick z axis input and therefore. determine the turn radius RCL as a function of reference pivot angle CL The. nomenclature is adopted here because it is an additional steering change on top. of the Crab steering angle,FIGURE 5 Twist 1 Geometry. CL is assigned proportionally from the joystick z axis input For the. sake of simplicity I limited the maximum CL values to the range 4. This limitation keeps the turning centerpoint from coming inside the. chassis circle In Snake Mode the limitation is 2 The chassis turn. radius RCP is calculated,A turn radius for each wheel Ri can be calculated. Ri h 2 cos 2 i RCP h sin i, As well as a twist steering angle i Note the following equation may.
need to be negated or otherwise modified based on joystick input specs. i joystick z sign sin 1 eq 6, The wheel drive velocity factor for each wheel is the ratio. where Rmax is the greatest turning radius for the wheels. A worksheet for Twist 1 calculations was developed. The most significant complication of Twist 1 as compared to Snake Mode. is that the calculations depend upon the chassis orientation and can. therefore not be easily pre calculated unless you parse and pre calculate. a 3 d array, Twist 1 is a static steering system Once the joystick controls are set all. steering and drive values are determined and do not change until the. joystick controls change,Math behind Twist 2, Twist 2 geometry is shown schematically in Figure 6 It s actually simpler. than Twist 1 It retains the useful concept of the h radius chassis circle. This chassis moves at a velocity V in s and rotates or twists at a rate of. radians s Note a capital V is used here for velocity to avoid. confusion with rotational rate in equations The imaginary Centerline. Wheel concept used in Twist 1 is not useful for Twist 2 We need look at. only one wheel i there are n wheels The wheel has a fixed orientation. relative to the chassis i same as Twist 1 is the chassis orientation. relative to the direction of travel and is now a function of t time where. 0 is a known initial condition Steering twist angle i is the. deviation from Crab Mode steering angle 2 is of course also. a function of t t,FIGURE 6 Twist 2 Geometry, Twist 2 rotational rate competes with chassis velocity V Any wheel. passing i 3 2 with a positive or i 2 with a negative must be. driven at a speed of V h Since the maximum drive speed is limited. to Vmax about 108 in s chassis and rotational velocities will need to be. balanced on the basis of y z axis Joystick inputs, The attached Model contains a reasonable allocation between chassis and.
rotational velocities At zero forward velocity and maximum z the. chassis will rotate at full drive speed At zero twist z and maximum. drive y the chassis will drive forward at full drive speed In the arena of. negotiation drive speed remains maximized at its peek and the twist and. forward drive play off against each other, The rotational position of the chassis can calculated. t 0 t eq 8, It is important to keep the value of between 0 and 2. To calculate the angular positions any for individual wheel i. i t i eq 9,As above keep these values between 0 and 2. x y and scalar wheel velocities are,Vx i t V h sin i eq 10. V y i t h cos i eq 11,Vi t V x2 i V y2 i eq 12,The calculation of i t is conditional.
If Vx i 0 then,i tan 1 eq 13a,If Vx i 0 and i 2 then. i tan 1 eq 13b,If Vx i 0 and i 2 then,i tan 1 eq 13c. There is a dimensionless number h V which characterizes the behavior. of Twist 2 If h V 1 the wheels do not pivot completely around with. each twist revolution they oscillate If on the other hand h V 1. then the wheels do pivot completely around with each twist revolution. These equations will work if V 0 In this case Twist 2 becomes. effectively Twist 3, A robust model has been developed based on 1st principles An earlier. model had been built from the back end forward but it broke under the. conditions h V 1 Under the conditions h V 1 the 1st principles. and back end models agree exactly The solution for the new model is. analytical The worksheet model is shown below, This is a dynamic system Setting the joystick position simply starts the. dynamic process Steering angles for the above twist are. And relative motor speeds, But each system of inputs provides a unique solution.
Pivot Wheel Drive Crab with a Twist Clem McKown Team 1640 13 November 2009 eq 1 edited 29 March 2010 4 Wheel Independent Pivot Wheel Drive describes a 4wd drive train in which each of the 4 wheels are independently driven and may be independently pivoted for steering purposes The design offers the potential for excellent drive

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