AXY:Element Coordination/One Dimensional Tracking Algorithm

One Dimensional Tracking Algorithm

This one us used always in simple axis based MF tracking:

The algorithm is based on a state machine:

State: Full Synchronization follow the master

• track ← master

• Check if master velocity or acceleration exceeds max master frame's values if yes return an error

• flopstime = flops = 0;

State: Tracking process

Initial Setup and testing

• Check if master velocity or acceleration exceeds max master frame's values if yes return an error
  

• Compute differences:
  

$pos_{d}iff=master_{p}os-track_{p}os$ pure delta-position
$vel_{d}iff=master_{v}el-track_{v}el$ Pure delta-velocity

Time-to-Reach estimation

This is an estimation of time needed to reach the target - it is not an exact formula if the velocity difference is positive, it means the robot velocit is lower then conveyors therefore we need more time once to get to a higher velocity then the conveyers and once to get from that (higher) velocity to conveyers - therefore the multiplication by 2

if ($vel_{d}iff>0$) Failed to parse (unknown function "\abs"): {\displaystyle time_to_reach = \sqrt(4*\abs(vel_diff)/sync_jrk) + \sqrt(4*\abs(pos_diff)/sync_acc)}
else Failed to parse (unknown function "\abs"): {\displaystyle time_to_reach = \sqrt(2*\abs(vel_diff)/sync_jrk) + \sqrt(4*\abs(pos_diff)/sync_acc)}

Additionaly we mutliply by a prediction factor - to decrease acc/jerk

time_to_reach *= PredictionFactor time_to_reach = T*int(time_to_reach/T+1) round to integer number of samples

Prediction of Randevous point

The predicted position of the randevous (minus T, if it will hapen in this sample!) double t = (time_to_reach-T); double pred_pos = master_pos + t*(master_vel + t*master_acc/2); // position prediction assuming constant accleration double pred_diff = pred_pos - track_pos; // Predicted position difference

double pred_vel = master_vel + t*master_acc; velocity prediction double pred_acc = 0; acceleration will be zero

Polynomial coefficients of 5-degree polynom connecting the current p,v,a to the randevous p,v,a $a0=track_{p}os$ - not used, directly written into code
$a1=track_{v}el$ - not used, directly written into code
$a2=0.5*track_{a}cc$ - not used, directly written into code
$a3=(0.5*(pred_{a}cc-3*track_{a}cc)+(10*pred_{d}iff/time_{t}o_{r}each-6*track_{v}el-4*pred_{v}el)/time_{t}o_{r}each)/time_{t}o_{r}each$
$a4=(((5*pred_{d}iff/time_{t}o_{r}each-(pred_{v}el+4*track_{v}el))/time_{t}o_{r}each-1.5*track_{a}cc)/time_{t}o_{r}each-2*a3)/time_{t}o_{r}each$
$a5=((((pred_{d}iff/time_{t}o_{r}each-track_{v}el)/time_{t}o_{r}each-0.5*track_{a}cc)/time_{t}o_{r}each-a3)/time_{t}o_{r}each-a4)/time_{t}o_{r}each$

Next Sample Position

The position at the next sample will be: dp = ((((a5*T + a4)*T + a3)*T + 0.5*track_acc)*T + track_vel)*T - Initial increment

The accleration and jerk increments could be computed in this way also: .... volatile double ddp = (((5*4*a5*T + 4*3*a4)*T + 3*2*a3)*T + track_acc)*T*T; // a*T*T volatile double dddp = ((5*4*3*a5*T + 4*3*2*a4)*T + 3*2*a3)*T*T*T*T; // j*T*T*T But they are computed as average values for the time of sample duration, this approach doeas fit more to the tools we are using to test this feature, (BMDS...) as they all compute accleration & jerk by numeric differentation inside one sample. The side effect of all this is that we can have a change in jerk/acc sign inside one sample and the average value we are using ddp,dddp could have an opposite sign, so we are seeing effects like jerk or accleeration is limited from a "wrong side" i.e. instead using jmax -jmax is used. There is not much we can do against it, using exact jerk/acc values (from polynom) returns than wrong avergae values we observe as jerk/acc exceed over theri max values. So the avergae value approach is not perefect but gives results within the given limits.

T1 = T limit = false if the filter did work - this flag will be on state.satVel = state.satAcc = state.satJrk = false;

FILTER

if (dp > sync_velT)

dp =  sync_velT; 
limit = state.satVel = true; 

else if (dp < -sync_velT)

dp = -sync_velT; 
limit = state.satVel = true;

ddp = dp - dp0

if (ddp > sync_accT2)

	
dp  =  sync_accT2 + dp0;	
ddp =  sync_accT2;  
limit = state.satAcc = true;

else if (ddp < -sync_accT2)

	
dp  = -sync_accT2 + dp0;	
ddp = -sync_accT2;  
limit = state.satAcc = true;

dddp = ddp - ddp0; if (dddp > sync_jrkT3 )

 
ddp =  sync_jrkT3 + ddp0; 
dp = ddp + dp0;  
limit = state.satJrk = true;

else if (dddp < -sync_jrkT3 )

 
ddp = -sync_jrkT3 + ddp0; 
dp = ddp + dp0;  
limit = state.satJrk = true;

track_pos += dp; // new position

if(limit) if the velocity,accleration or jerk is limited then the velocity computed using polynomials does not fit any more, this is observable as PCMD/VCDM missmatch (Issue 2596) </pre> track_vel = dp/T; track_acc = ddp/T/T;

else
'' Velocity & Accleration will be computed exactly not using dp and ddp, it has been shown that this''
'' approach is much better - means more stable''
<pre>
track_vel   = (((5*a5*T1  + 4*a4)*T1  + 3*a3)*T1  + track_acc)*T1 + track_vel;
track_acc   = ((20*a5*T1 + 12*a4)*T1  + 6*a3)*T1   + track_acc;
<pre>
''keep the old values for next sample''
ddp0 = ddp; 
dp0  = dp;

''System is synchronized if NEW delta's are under certesian thresholds:''
dv0 = dv; // keep the previous dv
dv = master_vel - track_vel;
if((fabs(dv) <= syncGain * sync_accT) && (fabs(master_pos - track_pos) <= syncGain * sync_accT2))  
'' Synchronization''
'' Take the middle for an additional smoothing:''
<pre>
track_pos	   =	0.5*(track_pos + master_pos);
track_vel	   =	0.5*(track_vel + master_vel);
ResetFilter();		// Reset the filter
state.tracking 	= MOT_MASTER_SYNC; // System synchronized

else if( dv0*dv < 0)

if (flops == -1) flops = 0; // ignore the first sample
else	  			  flops++;
flopstime = 0;

if (flops > maxFlops && (flopstime > maxFlopsTime && fabs(dv) > syncGain * sync_accT2))

state.tracking 	= MOT_MASTER_OSC; // OScilationss

else

flopstime++;

State: Stopping (de-synchronization process)

flopstime = flops = 0;

• De-Syncing profile, follow the given profile, ignore the real source

desync.Profile();
track_pos = track_pos_0+master_direction*desync.path.curr_pos;   	// position
track_vel = master_direction*desync.path.vel;			// velocity
track_acc = master_direction*desync.path.acc;			// acceleration
if(desync.path.Status.stop)			// if stopped,change the state to OFF
	 SetState(MOT_MASTER_OFF);