[Planning] what is the difference between PIECEWISE_JERK_SPEED_OPTIMIZER and PIECEWISE_JERK_NONLINEAR_SPEED_OPTIMIZER

[qwetqwe]

As I see in lane_follow_config, there are two tasks PIECEWISE_JERK_SPEED_OPTIMIZER and PIECEWISE_JERK_NONLINEAR_SPEED_OPTIMIZER. It seems that they both use QP to smooth the st path,what is the difference between them?

[zerolover]

PIECEWISE_JERK_SPEED_OPTIMIZER is formulated as a QP problem. PIECEWISE_JERK_NONLINEAR_SPEED_OPTIMIZER is formulated as a NLP(non linear program) problem since it considered the smoothness of Centripetal Acceleration （it's a non linear objective function）meanwhile. You can refer to code or this paper for more details.

Zhang, Yajia, et al. "Optimal Trajectory Generation for Autonomous Vehicles Under Centripetal Acceleration Constraints for In-lane Driving Scenarios." 2019 IEEE Intelligent Transportation Systems Conference (ITSC). IEEE, 2019.

[qwetqwe]

@zerolover ,thanks. I have another question. Before using NLP optimize, it uses QP optimize to smooth the curvature first.But I think curvature is related with path geometry. It only changes curvature and path point does not change. So what is the reason? here is the code

[zerolover]

You can read the paper first. In short, the NLP required the kappa should be differentiable, so the code here is to compute a curve "kappa(s)" which has constant jerk (you can also use other algothrim to fit the curve).

[qwetqwe]

@zerolover ,But the path from PIECEWISE_JERK_PATH_OPTIMIZER has been smoothed, so the curvature should be differentiable.Here smoothing again is for redundancy?

[zerolover]

We want to get kappa(s), a close form and smooth curve, hence we can get the 1st and 2nd derivative of kappa. See piecewise_jerk_speed_nonlinear_ipopt_interface.cc

[qwetqwe]

@zerolover thanks, you really help me a lot. Another question,do you know why they use soft safety bounds? The paper did not mention that. they add two slack variables to state variable and give them variable bounds: lower_s_slack[i]>=0 upper_s_slack[i]>=0 the objection function is: f=lower_s_slack[i]w_softbounds+upper_s_slack[i]w_softbounds and add some constraints: s[i] - soft_lower_s[i] + lower_slack[i] >= 0.0 s[i] - soft_upper_s[i] - upper_slack[i] <= 0.0 I think it is equal to s[i] - soft_lower_s[i]>=0 s[i] - soft_upper_s[i] <=0 why not add s bounds directly like: s[i]>=soft_lower_s[i] s[i]<=soft_upper_s[i]

[LiJiangnanBit]

@qwetqwe s[i] - soft_lower_s[i] + lower_slack[i] >= 0.0 s[i] - soft_upper_s[i] - upper_slack[i] <= 0.0

These are two soft constraits on s[i], which means it is possible that s[i] < soft_lower_s[i] or s[i] > soft_upper_s[i], if the slack variable lower_slack[i] or upper_slack[i] is positive enough. However, this would be punished by the cost function.

It expects s[i] to be within the range between soft_lower_s[i] and soft_upper_s[i], but it's ok to go beyond that if necessary.

[qwetqwe]

@LiJiangnanBit , I understand it, soft constraints means a buffer. thanks a lot

[lucianzhong]

@zerolover Could you please send me pdf of the paper "Optimal Trajectory Generation for Autonomous Vehicles Under Centripetal Acceleration Constraints for In-lane Driving Scenarios", can not find one on internet. My email is 373605178@qq.com Thanks a lot

[zerolover]

@lucianzhong use sci-hub to search the paper :)

[lucianzhong]

@zerolover Got it. Thanks a lot

[breezygogo]

@zerolover ,But the path from PIECEWISE_JERK_PATH_OPTIMIZER has been smoothed, so the curvature should be differentiable.Here smoothing again is for redundancy? path is using piecewise to generate so dkappa is 0,so use piecewise to fit not smoothed just fit。