Path Control in Robotics

Path Control in RoboticsME 4135Richard R. Lindeke, Ph. D.Motion Types of InterestPoint to Point Motion:All Axes start and end simultaneouslyAll Geometry is computed for targets and relevant Joint changes which are then forced to be followed during program executionPath or Trajectory Controller MotionHere the motion is performed through a time sequence of intermediate configurations computed ahead of time (like above but without stop-start operation) or in real timePaths are Space Curves for the n-Frame to followThis motion is a continuous scheme to move the TCP from one location to the next along a desired (straight or curved) line under direct operational control

Path Control and Motion Types:We will explore the following types of Motion:Lead Through Path Creation(Cubic) Polynomial Paths w/ Via PointsMinimum Time Trajectory w/ controlled AccelerationLower order Path-Poly ControlLSPB PathsCraigs Method for acceleration smoothingStrict Velocity ControlJoint Interpolated ControlFull Cartesian ControlLead Through Path CreationBasically this was a technique whereby a skilled operator took a robot arm (for welding or painting) and used it like his/her weld tool or paint sprayer and performed the required process at reasonable speed

The robot is equipped with a position recording device and memorizes a large number of points during the teaching sessionThese learned points then would be played back to replicate the skilled operators motions

Lead Through Path CreationAdvantages:Simple way to create complex pathsAll points are sure to be physically attainablePlayback speed can be controlled by an external deviceDisadvantages:Precision placements are required (program must be replayed at exactly the initial placement)Major concern with operator safety: robot is powered and operator is physically touching it (OSHA rules it unsafe practice!)Modern Path Control:(Lets look at a simple example)Dr. Ds new Self-powered Automated Coffee Drinker RobotIt is a simple cantilevered Cartesian device equipped with a spherical wrist that responds to eye movement and thoughts to help the overworked design engineer get coffee while designing and drafting & typing of Reports It follows a straight line path from the cups point on a table to the workers mouth in secondLets look at a simple example:

Lets look at a simple example:We see that the Bot must travel a space path of 16.45 which can be decomposed into a movement of 9.5 along each of the prismatic joints

For accuracy lets divide each of the these joint paths into 100 segments

From Physics: Vjoint d/ t = (9.5/100)/(.5/100) = 19in/second (a reasonable speed!)

Lets look at a simple example:During the 1st Step then:Joint 1 starts at 0 and moves to 0.095Moves there in 0.005 secondsHow will it do it?Of course by Accelerating from a stop to 19 in/sec in 0.005 secondsCompute Accreqr V/tDarn this says that the acceleration is 3800in/sec2 this is 10G!!!!!Lets look at a simple example:So this will certainly be difficult to accomplish! (more likely it will not work)OSHA would be just as upset as when we had the worker holding on to the powered robot what should we do?I think our approach is too nave!If we examine the Pos vs. Time, Vel Vs. Time and Acc vs. Time plots we may see why: Look at a simple examples Trajectory Curves:

This is Physically Impossible (or rather very energy intensive)Can we build a reasonable solution that keeps the acceleration to an achievable level?What this would mean is we wouldnt instantly in one time step go from stopped to full speedThis can be achieved with a time polynomial model of motionBuilding a Path Polynomial Motion Set

These are the trajectory equations for a joint (Position, Velocity and Acceleration)Solving the Path Polynomial finding ais for SPECIFIC PATHSWe would have boundary conditions for position and velocity at both ends of the pathWe would have the desired total time of travelUsing these conditions we can solve for a0, a1, a2 and a3 to build a 3rd order path polynomial for the required motionSolving the Path Polynomial finding ais for SPECIFIC PATHS

Polys holding at starting time and positionPolys holding at ending time and positionSolving the Path Polynomial finding ais for SPECIFIC PATHSWriting these as Matrix Forms:

Solving the Path Polynomial finding ais for SPECIFIC PATHSIf we set t0 = 0 (starting time is when we start counting motion!) then:

By examination, a0 = q0 & a1 = q0(dot)Solving the Path Polynomial finding ais for SPECIFIC PATHSCompleting the solution consists of forming relationships for: a2 & a3Done by substituting a0 & a1 values and solving the last two equation simultaneously:

Be Careful and note the order of the positions and velocities!Applying it to the Coffee BotStart: X = 0; v = 0 @ time = 0End: X = 9.5; v = 0 @ time = .5 seca0 = 0 ; a1 = 0a2 = (3 * 9.5)/(0.52) = 114a3 = (2 *(- 9.5))/(0.53) = -152Applying it to the Coffee BotHere (specifically):

Applying it to the Coffee BotSimplifying:

Applying it to the Coffee Bot: Position

Position vs. TimeApplying it to the Coffee Bot: Velocity

Joint Velocity Vs. TimeApplying it to the Coffee Bot: Acceleration

Acceleration Vs. TimeApplying it to the Coffee BotUsing the Path Polynomial Approach:Max Velocity is: 28.5 in/sec (compares to 19 in/sec)Max Acceleration is: 228 in/sec2 (.6 g) compared to 10 gBut, in this method, I require a 100% duty cycle motor since throughout the entire path, the motor is accelerating (either with positive or negative orientation)Can we make a path solution where we accelerate for only part of the path? Turns out we can and we will call it LSPB!Studying the LSPB modelIn this model, we will carry forward with a parabolic model that is 2nd orderIn this model, we will determine a time where we will blend from startup until we reach a constant velocity (and its greater than 1/100 of the total time!)Here we will see an acceleration followed by a period of coasting and then deceleration (often called a trapezoidal velocity model)

Model Building:We must define an acceleration constraint (minimum value) such that the acceleration is guaranteed to be completed within half of the allocated time period of the travel:

This assures that there is no overlap for the BLEND RegionsLooking at the motion over the various regions:During Region 1 (while the joint is Accelerating) (time interval 0 to tblend [tb]) the Joint moves: q = (V/2)*tbtb is the acceleration timeDuring the region of Constant Velocity the Joint moves: q = V*(t 2tb)During Region 3 -- while the joint is decelerating the joint moves: q = (V/2)*tbTotal travel distance is qB - qA

Writing a motion equation:

Substituting and Isolating the Unknown (blend time):

This is a quadratic equation in tbSolving for tb:

Note: Acceleration is subject to constraint determined aboveApplying it to the Coffee BotAcceleration constraint:

Blend time:

Applying it to the Coffee Bottlinear=.5 2*0.159 = 0.181 s

Linear Velocity:

Positions:By tb, the Joint has moved: (27.89/2)*.159 = 2.222During linear velocity joint moves: 0.181*27.89 = 5.055 in (thus the pos = 7.277)During deceleration joint travels 2.222Adding them gives full travel distance: 2.222 + 5.055 + 2.222 in 9.5in

Plotting the Path trajectory:

Notice: the accelerator is off during the linear travel segment The 2 Previous Path Control Methods focused on Start/Stopping ApproachesWhat can we do if we desire to travel by continuing along a path w/o stopping at each point?

Here we will focus on a method called dog-tracking after the lead and follow techniques employed in dog racing

Essentially we would have a situation where the path is laid out (as a series of Via Points) and the joints smoothly maneuver through and between themCraigs Dog Tracking Method

Craigs Dog Tracking MethodUpon Examination of the motion, we find that there are three regimes in the motionThese are:Start up regimeIntermediate regimesStopping regimeStarting and Stopping are similar to LSPB in the way they compute blend time and accelerationDuring an Intermediate regime we compute acceleration by comparing incoming and outgoing velocities about each pointCraigs Dog Tracking MethodStarting regime Equations:

Start AccelerationStart Blend TimeLinear Velocity 12Time @ linear VelocityCraigs Dog Tracking MethodStopping Equations:

Stop accelerationStop Blend TimeL. Velocity to stopTime @ L. VelocityCraigs Dog Tracking MethodIntermediate Equations:

Linear VelocityAccelerationBlend timeTime @ L. VelocityCraigs Dog Tracking MethodUpon examination of the set of equation on the previous 3 slides several point should be noted: Start and Stop are essentially the same but very important differences must be notedOne cant complete any of the regimes without looking ahead actually looking ahead to the 2nd point beyond to see if a joint is stopping or continuingStart/Stop require position/time relationshipsIntermediate regimes require velocity/time relationships Craigs Dog Tracking MethodStep 1: Calculate Global usable acceleration (magnitude) constraint based on LSPB model applied Pairwise (12; 23; etc) Step 2: Focus on Start and Stop Segments

Step 3: Complete the table of accelerations, blend times, linear velocity and time at linear velocity

Point NamePosition (inch)Delta Position (inch)Arrival Time (sec)Delta Time (sec)A00-------B9.59.50.5.5C111.53.53D9.5-1.56.53E0-9.57.51Lets Expand on Dr. Ds Coffee Bot:Lets Expand on Dr. Ds Coffee Bot:Step 1: Global Acc. Constraint

This is largest should work globally but lets make sure it doesnt miss so choose 200ips2 cause its easier to calculate and is only about .6gLets Expand on Dr. Ds Coffee Bot:Next we focus on the start & stop equations:Starting

Stopping

NOTE: cant compute tlij yet we lack the data!Lets Expand on Dr. Ds Coffee Bot:Considering Intermediate BC:

Lets Expand on Dr. Ds Coffee Bot:On to CD Segment

Lets Expand on Dr. Ds Coffee Bot:Now for Segment DE:

SummarizingPtPOST. TimePOStimeTi (blend)AccL. VelTime@L. VelA00---------.106+20021.3.342B9.5.59.5.5.103-200.52.946C113.51.53.005-200-.52.974D9.56.5-1.53.046-200-9.74.928E07.5-9.51.049+200A Final Thought on Dog-Tracking:

But if we must travel over or through a certain point, we can define Pseudo-Via points that flank the desired target and force the arm to pass the pseudos and drive right over the original desired target pointLooking at Velocity Control

Achievable PathDesired PathWith this Velocity ControlThe acceleration is set and the new velocity for an upcoming segment is inserted at the appropriate time (place)

Over the segment, the arm (joints) lags the desired path

We extend the next path velocity curve to intersect the actual achievable path (to a point that is earlier in time than when we would have expected to change) this will mean that the joint can catch up to the desired plan for the travel as it blends to the new velocity

Linear Path ControlSometimes thought of as Cartesian ControlIt is based on the idea of Transitions between consecutive required geometriesThese transitions are based on the solution of a Drive Matrix:

The matrices T(r), Ra(r), and Ro(r) are the translation, rotation wrt Z and rotation wrt Y in transitioning from Pinitial to PfinalDeveloping the Drive Matrix:Given: P1 is (n1, o1, a1, d1)And: P2 is (n2, o2, a2, d2)Then:

Cartesian ControlIt is used when very exact interaction is requiredIt guaranties accurate tool placement at all timesIt is typically used in time dependent solutions like interaction while a product is moving

Cartesian Control

NOTE: On the conveyor, the H-frame is a time dependent pose in C (conveyor space)Cartesian ControlWe desire to attach the Quality Tag to the part as it moves by the robot stationRequires that the part and robot tool must be in exact contact throughout the attachment processThis becomes a Time-based Mapping problemCartesian ControlAt Time 1 (P1):

At Time 2 (P2):

Cartesian ControlUsing these two (time-dependent) Poses, we can build the desired drive matrixWe can compute the accuracy of the path then as a series of changes to the three control vectors: a, o and dThese are updated in real timeCartesian ControlProblems that can result (and must be accounted for):Intermediate points that are unreachable After we compute the initial and final points (that prove to be reachable as individuals), we request the tracking of a, o and d vectors but they exceed joint capabilities or require positions outside the work envelope during the driving actionIn certain situations where only certain solutions are possible for the robot, like being near singularities, the desired linear velocity may require very high joint velocities exceeding capabilities and the path actually followed will deviate from the one desired as the joints run at their velocity limits Near Cartesian (Joint Interpolated) ControlThis is a semi-precise control method developed as a compromise between full-Cartesian and point-to-point motionBasically it is used when a process needs to be held within a band about an ideal linear path for example during painting or bar-code scanningThe path is designed to track the work as it moves and maintains no more than a given focal distance separation between the tool and work surfaceIt is a path that is close to the target path at all poses but exact only at a few!

Joint-Interpolated ControlStep 1: determine the desired pathStep 2: Compute the tolerable error and the number of points (VIAs) needed to maintain tooltowork distancesStep 3:Compute IKSs at each of the VIAsStep 4: Determine Move time for each segment:

Joint-Interpolated ControlStep 5: Divide the Tseg into m equal time intervals:

Joint-Interpolated ControlStep 6: For each joint, determine angular distance during each time segment tseg:

Step 7: at the beginning of the nth step over a path, joint i servo control receives a target point:

Joint-Interpolated ControlImplementing this method begins with determination of the distance between and ultimately the number of Via Points neededThis is (really!) a simple trigonometry problem based on the offset distance and error tolerance () at closest approach

Joint-Interpolated Control -- Model

Robot Base212Note: R = R1 = R2Joint-Interpolated Control -- ModelNotice line #1, #2 and R1 form a right triangle

#2 is half the distance between Via Points!R (= R1) is measured at point of closest approach between the robot and part!Lets try one:A Part 6m long moves by a stationary robot on a conveyor moving at 0.04mps (counter flowing compared to painting direction) If we desire that the robot complete its spraying in 15 seconds Then, the robot must travel 5.4 m to spray the side of the part nearest it since the part moves during the painting operation.At closest approach, the robot is 1.5 m from the part and needs to have its sprayer 20 cm ( 5 cm) from the part.From this data, the R value is: 1.5 (.20 + .05) = 1.35 m

Lets try one:Distance between Vias is found using:

Therefore: The number of Vias = 5.4/1.020 = 5.29 so round up to 6plus the initial point = 6 + 1 = 7

Follow-upDistance between is: 5.4/6 = .9m (rather than 1.020m)What is actual Error band?Here we see it is: 3.86cmTypically, we find that Joint Interpolated solutions provide better than required (or expected) process control!

What if we equally space the Vias?Chart100.0028310.0112480.0251370.0443840.0688750.0984960.1331330.1726720.2169990.2660.3195610.3775680.4399070.5064640.5771250.6517760.7303030.8125920.8985290.9881.0808911.1770881.2764771.3789441.4843751.5926561.7036731.8173121.9334592.0522.1728212.2958082.4208472.5478242.6766252.8071362.9392433.0728323.2077893.3443.4813513.6197283.7590173.8991044.0398754.1812164.3230134.4651524.6075194.754.8924815.0348485.1769875.3187845.4601255.6008965.7409835.8802726.0186496.1566.2922116.4271686.5607576.6928646.8233756.9521767.0791537.2041927.3271797.4487.5665417.6826887.7963277.9073448.0156258.1210568.2235238.3229128.4191098.5128.6014718.6874088.7696978.8482248.9228758.9935369.0600939.1224329.1804399.2349.2830019.3273289.3668679.4015049.4311259.4556169.4748639.4887529.4971699.5

Postime (sec)Pos of Joint (in)

Chart201.12862.23443.31744.37765.4156.42967.42148.39049.336610.2611.160612.038412.893413.725614.53515.321616.085416.826417.544618.2418.912619.562420.189420.793621.37521.933622.469422.982423.472623.9424.384624.806425.205425.581625.93526.265626.573426.858427.120627.3627.576627.770427.941428.089628.21528.317628.397428.454428.488628.528.488628.454428.397428.317628.21528.089627.941427.770427.576627.3627.120626.858426.573426.265625.93525.581625.205424.806424.384623.9423.472622.982422.469421.933621.37520.793620.189419.562418.912618.2417.544616.826416.085415.321614.53513.725612.893412.038411.160610.269.33668.39047.42146.42965.4154.37763.31742.23441.12860

VelTime (sec)Velocity (in/sec)

Sheet1PosVelAcc0002280.0050.0028311.1286223.440.010.0112482.2344218.880.0150.0251373.3174214.320.020.0443844.3776209.760.0250.0688755.415205.20.030.0984966.4296200.640.0350.1331337.4214196.080.040.1726728.3904191.520.0450.2169999.3366186.960.050.26610.26182.40.0550.31956111.1606177.840.060.37756812.0384173.280.0650.43990712.8934168.720.070.50646413.7256164.160.0750.57712514.535159.60.080.65177615.3216155.040.0850.73030316.0854150.480.090.81259216.8264145.920.0950.89852917.5446141.360.10.98818.24136.80.1051.08089118.9126132.240.111.17708819.5624127.680.1151.27647720.1894123.120.121.37894420.7936118.560.1251.48437521.3751140.131.59265621.9336109.440.1351.70367322.4694104.880.141.81731222.9824100.320.1451.93345923.472695.760.152.05223.9491.20.1552.17282124.384686.640.162.29580824.806482.080.1652.42084725.205477.520.172.54782425.581672.960.1752.67662525.93568.40.182.80713626.265663.840.1852.93924326.573459.280.193.07283226.858454.720.1953.20778927.120650.160.23.34427.3645.60.2053.48135127.576641.040.213.61972827.770436.480.2153.75901727.941431.920.223.89910428.089627.360.2254.03987528.21522.80.234.18121628.317618.240.2354.32301328.397413.680.244.46515228.45449.120.2454.60751928.48864.560.254.7528.500.2554.89248128.4886-4.560.265.03484828.4544-9.120.2655.17698728.3974-13.680.275.31878428.3176-18.240.2755.46012528.215-22.80.285.60089628.0896-27.360.2855.74098327.9414-31.920.295.88027227.7704-36.480.2956.01864927.5766-41.040.36.15627.36-45.60.3056.29221127.1206-50.160.316.42716826.8584-54.720.3156.56075726.5734-59.280.326.69286426.2656-63.840.3256.82337525.935-68.40.336.95217625.5816-72.960.3357.07915325.2054-77.520.347.20419224.8064-82.080.3457.32717924.3846-86.640.357.44823.94-91.20.3557.56654123.4726-95.760.367.68268822.9824-100.320.3657.79632722.4694-104.880.377.90734421.9336-109.440.3758.01562521.375-1140.388.12105620.7936-118.560.3858.22352320.1894-123.120.398.32291219.5624-127.680.3958.41910918.9126-132.240.48.51218.24-136.80.4058.60147117.5446-141.360.418.68740816.8264-145.920.4158.76969716.0854-150.480.428.84822415.3216-155.040.4258.92287514.535-159.60.438.99353613.7256-164.160.4359.06009312.8934-168.720.449.12243212.0384-173.280.4459.18043911.1606-177.840.459.23410.26-182.40.4559.2830019.3366-186.960.469.3273288.3904-191.520.4659.3668677.4214-196.080.479.4015046.4296-200.640.4759.4311255.415-205.20.489.4556164.3776-209.760.4859.4748633.3174-214.320.499.4887522.2344-218.880.4959.4971691.1286-223.440.59.50-228

Sheet2

Sheet3

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Postime (sec)Pos of Joint (in)

Chart201.12862.23443.31744.37765.4156.42967.42148.39049.336610.2611.160612.038412.893413.725614.53515.321616.085416.826417.544618.2418.912619.562420.189420.793621.37521.933622.469422.982423.472623.9424.384624.806425.205425.581625.93526.265626.573426.858427.120627.3627.576627.770427.941428.089628.21528.317628.397428.454428.488628.528.488628.454428.397428.317628.21528.089627.941427.770427.576627.3627.120626.858426.573426.265625.93525.581625.205424.806424.384623.9423.472622.982422.469421.933621.37520.793620.189419.562418.912618.2417.544616.826416.085415.321614.53513.725612.893412.038411.160610.269.33668.39047.42146.42965.4154.37763.31742.23441.12860

VelTime (sec)Velocity (in/sec)

Sheet1PosVelAcc0002280.0050.0028311.1286223.440.010.0112482.2344218.880.0150.0251373.3174214.320.020.0443844.3776209.760.0250.0688755.415205.20.030.0984966.4296200.640.0350.1331337.4214196.080.040.1726728.3904191.520.0450.2169999.3366186.960.050.26610.26182.40.0550.31956111.1606177.840.060.37756812.0384173.280.0650.43990712.8934168.720.070.50646413.7256164.160.0750.57712514.535159.60.080.65177615.3216155.040.0850.73030316.0854150.480.090.81259216.8264145.920.0950.89852917.5446141.360.10.98818.24136.80.1051.08089118.9126132.240.111.17708819.5624127.680.1151.27647720.1894123.120.121.37894420.7936118.560.1251.48437521.3751140.131.59265621.9336109.440.1351.70367322.4694104.880.141.81731222.9824100.320.1451.93345923.472695.760.152.05223.9491.20.1552.17282124.384686.640.162.29580824.806482.080.1652.42084725.205477.520.172.54782425.581672.960.1752.67662525.93568.40.182.80713626.265663.840.1852.93924326.573459.280.193.07283226.858454.720.1953.20778927.120650.160.23.34427.3645.60.2053.48135127.576641.040.213.61972827.770436.480.2153.75901727.941431.920.223.89910428.089627.360.2254.03987528.21522.80.234.18121628.317618.240.2354.32301328.397413.680.244.46515228.45449.120.2454.60751928.48864.560.254.7528.500.2554.89248128.4886-4.560.265.03484828.4544-9.120.2655.17698728.3974-13.680.275.31878428.3176-18.240.2755.46012528.215-22.80.285.60089628.0896-27.360.2855.74098327.9414-31.920.295.88027227.7704-36.480.2956.01864927.5766-41.040.36.15627.36-45.60.3056.29221127.1206-50.160.316.42716826.8584-54.720.3156.56075726.5734-59.280.326.69286426.2656-63.840.3256.82337525.935-68.40.336.95217625.5816-72.960.3357.07915325.2054-77.520.347.20419224.8064-82.080.3457.32717924.3846-86.640.357.44823.94-91.20.3557.56654123.4726-95.760.367.68268822.9824-100.320.3657.79632722.4694-104.880.377.90734421.9336-109.440.3758.01562521.375-1140.388.12105620.7936-118.560.3858.22352320.1894-123.120.398.32291219.5624-127.680.3958.41910918.9126-132.240.48.51218.24-136.80.4058.60147117.5446-141.360.418.68740816.8264-145.920.4158.76969716.0854-150.480.428.84822415.3216-155.040.4258.92287514.535-159.60.438.99353613.7256-164.160.4359.06009312.8934-168.720.449.12243212.0384-173.280.4459.18043911.1606-177.840.459.23410.26-182.40.4559.2830019.3366-186.960.469.3273288.3904-191.520.4659.3668677.4214-196.080.479.4015046.4296-200.640.4759.4311255.415-205.20.489.4556164.3776-209.760.4859.4748633.3174-214.320.499.4887522.2344-218.880.4959.4971691.1286-223.440.59.50-228

Sheet2

Sheet3

Chart100.0028310.0112480.0251370.0443840.0688750.0984960.1331330.1726720.2169990.2660.3195610.3775680.4399070.5064640.5771250.6517760.7303030.8125920.8985290.9881.0808911.1770881.2764771.3789441.4843751.5926561.7036731.8173121.9334592.0522.1728212.2958082.4208472.5478242.6766252.8071362.9392433.0728323.2077893.3443.4813513.6197283.7590173.8991044.0398754.1812164.3230134.4651524.6075194.754.8924815.0348485.1769875.3187845.4601255.6008965.7409835.8802726.0186496.1566.2922116.4271686.5607576.6928646.8233756.9521767.0791537.2041927.3271797.4487.5665417.6826887.7963277.9073448.0156258.1210568.2235238.3229128.4191098.5128.6014718.6874088.7696978.8482248.9228758.9935369.0600939.1224329.1804399.2349.2830019.3273289.3668679.4015049.4311259.4556169.4748639.4887529.4971699.5

Postime (sec)Pos of Joint (in)

Chart201.12862.23443.31744.37765.4156.42967.42148.39049.336610.2611.160612.038412.893413.725614.53515.321616.085416.826417.544618.2418.912619.562420.189420.793621.37521.933622.469422.982423.472623.9424.384624.806425.205425.581625.93526.265626.573426.858427.120627.3627.576627.770427.941428.089628.21528.317628.397428.454428.488628.528.488628.454428.397428.317628.21528.089627.941427.770427.576627.3627.120626.858426.573426.265625.93525.581625.205424.806424.384623.9423.472622.982422.469421.933621.37520.793620.189419.562418.912618.2417.544616.826416.085415.321614.53513.725612.893412.038411.160610.269.33668.39047.42146.42965.4154.37763.31742.23441.12860

VelTime (sec)Velocity (in/sec)

Chart3228223.44218.88214.32209.76205.2200.64196.08191.52186.96182.4177.84173.28168.72164.16159.6155.04150.48145.92141.36136.8132.24127.68123.12118.56114109.44104.88100.3295.7691.286.6482.0877.5272.9668.463.8459.2854.7250.1645.641.0436.4831.9227.3622.818.2413.689.124.560-4.56-9.12-13.68-18.24-22.8-27.36-31.92-36.48-41.04-45.6-50.16-54.72-59.28-63.84-68.4-72.96-77.52-82.08-86.64-91.2-95.76-100.32-104.88-109.44-114-118.56-123.12-127.68-132.24-136.8-141.36-145.92-150.48-155.04-159.6-164.16-168.72-173.28-177.84-182.4-186.96-191.52-196.08-200.64-205.2-209.76-214.32-218.88-223.44-228

Acctime (sec)Acceleration (in/sec^2)

Sheet1PosVelAcc0002280.0050.0028311.1286223.440.010.0112482.2344218.880.0150.0251373.3174214.320.020.0443844.3776209.760.0250.0688755.415205.20.030.0984966.4296200.640.0350.1331337.4214196.080.040.1726728.3904191.520.0450.2169999.3366186.960.050.26610.26182.40.0550.31956111.1606177.840.060.37756812.0384173.280.0650.43990712.8934168.720.070.50646413.7256164.160.0750.57712514.535159.60.080.65177615.3216155.040.0850.73030316.0854150.480.090.81259216.8264145.920.0950.89852917.5446141.360.10.98818.24136.80.1051.08089118.9126132.240.111.17708819.5624127.680.1151.27647720.1894123.120.121.37894420.7936118.560.1251.48437521.3751140.131.59265621.9336109.440.1351.70367322.4694104.880.141.81731222.9824100.320.1451.93345923.472695.760.152.05223.9491.20.1552.17282124.384686.640.162.29580824.806482.080.1652.42084725.205477.520.172.54782425.581672.960.1752.67662525.93568.40.182.80713626.265663.840.1852.93924326.573459.280.193.07283226.858454.720.1953.20778927.120650.160.23.34427.3645.60.2053.48135127.576641.040.213.61972827.770436.480.2153.75901727.941431.920.223.89910428.089627.360.2254.03987528.21522.80.234.18121628.317618.240.2354.32301328.397413.680.244.46515228.45449.120.2454.60751928.48864.560.254.7528.500.2554.89248128.4886-4.560.265.03484828.4544-9.120.2655.17698728.3974-13.680.275.31878428.3176-18.240.2755.46012528.215-22.80.285.60089628.0896-27.360.2855.74098327.9414-31.920.295.88027227.7704-36.480.2956.01864927.5766-41.040.36.15627.36-45.60.3056.29221127.1206-50.160.316.42716826.8584-54.720.3156.56075726.5734-59.280.326.69286426.2656-63.840.3256.82337525.935-68.40.336.95217625.5816-72.960.3357.07915325.2054-77.520.347.20419224.8064-82.080.3457.32717924.3846-86.640.357.44823.94-91.20.3557.56654123.4726-95.760.367.68268822.9824-100.320.3657.79632722.4694-104.880.377.90734421.9336-109.440.3758.01562521.375-1140.388.12105620.7936-118.560.3858.22352320.1894-123.120.398.32291219.5624-127.680.3958.41910918.9126-132.240.48.51218.24-136.80.4058.60147117.5446-141.360.418.68740816.8264-145.920.4158.76969716.0854-150.480.428.84822415.3216-155.040.4258.92287514.535-159.60.438.99353613.7256-164.160.4359.06009312.8934-168.720.449.12243212.0384-173.280.4459.18043911.1606-177.840.459.23410.26-182.40.4559.2830019.3366-186.960.469.3273288.3904-191.520.4659.3668677.4214-196.080.479.4015046.4296-200.640.4759.4311255.415-205.20.489.4556164.3776-209.760.4859.4748633.3174-214.320.499.4887522.2344-218.880.4959.4971691.1286-223.440.59.50-228

Sheet2

Sheet3