319 |
Appendix
320 |
321 |
322 | The following sections provide additional technical details that supplement our paper. Please consult our
323 | paper for the definition of the symbols used in the following sections.
324 |
325 |
326 | $
327 | \gdef\paramdim{M}
328 | \gdef\params{\theta}
329 | \gdef\particles{\Theta}
330 | \gdef\numparticles{N}
331 | \gdef\statedim{D}
332 | \gdef\statevec{\mathbf{s}}
333 | \gdef\controlvec{\mathbf{u}}
334 | \gdef\observationdim{N}
335 | \gdef\observationvec{\mathbf{x}}
336 | \gdef\timedim{T}
337 | \gdef\trajectory{\mathcal{X}}
338 | \gdef\trajectoryset{D_\trajectory}
339 | \gdef\trajectorysetsub[1]{D_{\trajectory,#1}}
340 | \gdef\particleset{\xi}
341 | \gdef\simu{^{\text{sim}}}
342 | \gdef\real{^{\text{real}}}
343 | \gdef\fsim{f_{\text{sim}}}
344 | \gdef\fobs{f_{\text{obs}}}
345 | \gdef\fstep{f_{\text{step}}}
346 | \gdef\pobs{p_{\text{obs}}}
347 |
348 | \gdef\xp{\params^\prime}
349 | \gdef\kxx{k(\params, \xp)}
350 | \gdef\logprob{\log \mathcal{L}(\params)}
351 | \gdef\logprobp{\log \mathcal{L}(\xp)}
352 |
353 | \gdef\gradx{\nabla_{\params}}
354 | \gdef\gradxp{\nabla_{\xp}}
355 | \gdef\gradxxp{\nabla_{\params\xp}}
356 |
357 | \gdef\exparams{\bar{\params}}
358 |
359 | \gdef\jointpos{\mathbf{q}}
360 | \gdef\jointvel{\mathbf{\dot{q}}}
361 | \gdef\jointacc{\mathbf{\ddot{q}}}
362 | \gdef\jointtorque{\tau}
363 | \gdef\externalforce{\mathbf{f}^{\text{ext}}}
364 | \gdef\timestep{\Delta t}
365 |
366 | \gdef\pdef{p_{\text{def}}}
367 | \gdef\plim{p_{\text{lim}}}
368 |
369 | \gdef\second{\text{s}}
370 | \gdef\kg{\text{kg}}
371 | \gdef\meter{\text{m}}
372 | \gdef\Hz{\text{Hz}}
373 | $
374 |
375 |
1 Technical Details
376 |
1.1 Initializing the Estimators
377 |
378 | For each experiment, we initialize the particles for the estimators via the deterministic Sobol sequence on
379 | the intervals specified through the parameter limits. Our code uses the Sobol sequence implementation
380 | from Burkardt [1] which is based on a Fortran77 implementation by Fox [2].
382 | For the MCMC methods that sample a single Markov chain, we used the center point between the parameter
383 | limits as initial guess.
384 |
385 |
386 |
1.2 Likelihood Model for Sets of Trajectories
387 |
388 |
389 | In this work we assume that trajectories may have been generated by a distribution over parameters.
390 | In the case of a replicable experimental setup, this could be a point distribution at the only true
391 | parameters.
392 | However, when trajectories are collected from multiple robots, or with slightly different experimental
393 | setups between experiments}, there may be a multimodal distribution over parameters which generated the set
394 | of trajectories.
395 |
396 |
397 | Note, that irrespective of the choice of likelihood function we do not make any assumption about the shape
398 | of the posterior distribution by leveraging SVGD which is a non-parametric inference algorithm. In
399 | trajectory space, the Gaussian likelihood function is a common choice as it corresponds to the typical
400 | least-squares estimation methodology. Other likelihood distributions may be integrated with our method,
401 | which we leave up to future work.
402 |
403 |
404 | The likelihood which we use is a mixture of equally-weighted Gaussians centered at each reference
405 | trajectory $\trajectory\real$:
406 | $$
407 | \begin{equation}
408 | p_{sum} (\trajectoryset\real | \params) = \sum_{\trajectory\real \in \trajectoryset\real}
409 | p_{ss}(\trajectory\real | \params).
410 | \end{equation}
411 | $$
412 |
413 | If we were to consider each trajectory as an independent sample from the same trajectory distribution (the
414 | product), the likelihood function would be
415 | $$\begin{equation}
416 | p_{product} (\trajectoryset\real | \params) = \prod_{\trajectory\real \in \trajectoryset\real}
417 | p_{ss}(\trajectory\real | \params).
418 | \end{equation}$$
419 |
420 | While both equations define the likelihood for
421 | a combination of single-shooting likelihood functions $p_{ss}$ for a set of real trajectories
422 | $\trajectoryset\real$, the same combination operators (sum or product) apply to the combination of
423 | multiple-shooting likelihood functions $p_{ms}$ analogously.
424 |
425 |
426 | The consequence of using these likelihoods can be seen in the following figure where
427 | $p_{product}$ shows the resulting posterior distribution (in parameter space) when
428 | treating a set of trajectories as independent and taking the product of their likelihoods
429 | (Equation 2), while Figure 1 (b) shows the result of treating
430 | them as a sum of Gaussian likelihoods (Equation 1).
431 | In Figure 1 (a) the posterior becomes the average of the two
432 | distributions since that
433 | is the most likely position that generated both of the distinct trajectories. In contrast, the posterior
434 | approximated by the same algorithm (CSVGD) but using the sum of Gaussian likelihoods, successfully captures
435 | the multimodality in the trajectory space since most particles have aligned near the two modes of the true
436 | distribution in parameter space.
437 |
438 |
439 |
440 |
441 |
442 |
443 |
444 |
(a) Product
445 |
446 |
447 |
448 |
(b) Sum
449 |
450 |
451 |
452 |
453 |
454 |
455 |
456 |
457 | Figure 1. Comparison of posterior parameter distributions obtained from fitting the
458 | parameters to two
459 | ground-truth trajectories generated from different link lengths of a simulated double pendulum
460 | (units of the axes in meters). The trajectories were 300 steps long (which corresponds to a length
461 | of 3s) and contain the 2 joint positions and 2 joint velocities of the uncontrolled
462 | double pendulum which starts from a zero-velocity initial configuration where the first angle is at
463 | $90^\circ$ (sideways) and the other at $0^\circ$. In (a), the product of the individual
464 | per-trajectory likelihoods is maximized (Equation 2). In (b), the sum
465 | of the likelihoods is maximized (Equation 1).
466 |
467 |
468 |
469 |
470 |
471 |
1.3 State and Parameter Normalization
472 |
473 |
474 | The parameters we are estimating are valid over only particular ranges of values. These ranges are often
475 | widely different - in the case of our real-world pendulum experiment, the center of mass of a link in a
476 | pendulum may be in the order of centimeters, while the angular velocity at the beginning of the recorded
477 | motion can reach values on the orders of meters per second. It is therefore important to scale the
478 | parameters to a common range to avoid any dimension to dominate smaller parameter ranges during the
479 | estimation.
480 |
481 |
482 | Similarly, the state dimensions are of different units - for example, we typically include velocities and
483 | positions in the recorded data over which we compute the likelihood. Therefore, we also normalize the range
484 | over the state dimensions. Given the state vector, respective parameter vector, $w$, we normalize each
485 | dimension $i$ by its statistical variance $\sigma^2$, i.e. $\frac{w_i}{\sigma_i^2}$.
486 |
487 |
488 |
1.4 KNN-based Approximation for KL Divergence
489 |
490 | In this work, we compare a set of parameter guesses (particles) to the ground-truth parameters, or a set of
491 | trajectories generated by simulating trajectories from each parameter in the particle distribution to a set
492 | of trajectories created on a physical system.
493 | To compare these distributions, we use the KL divergence to determine how the two distributions differ from
494 | each other.
495 | Formally, the KL divergence is the expected value of the log likelihood ratio between two distributions, and
496 | is an asymmetric divergence that does not satisfy the triangle inequality.
497 |
498 |
499 | The KL divergence is easily computable in the case of discrete distributions or simple parametric
500 | distributions, but is not easily calculable for samples from non-parametric distributions such as those over
501 | trajectories.
502 | Instead, we use an approximation to the KL divergence which uses the relative distances between samples in a
503 | set to estimate the KL divergence between particle distributions.
504 | This method has been used to compare particle distributions over robot poses to asses the performance of
505 | particle filter distributions [3].
506 | To estimate the KL divergence between particle distributions over trajectories $\trajectoryset^{p}$ and
507 | $\trajectoryset^{q}$ we adopt the formulation from [4,3]:
509 | $$
510 | \begin{equation}
511 | \tilde{d}_{KL} (\trajectoryset^{p} \parallel \trajectoryset^{q}) =
512 | \frac{\observationdim}{|\trajectoryset^p|} \sum_{i=1}^{|\trajectoryset^p|}
513 | \log\frac{\operatorname{KNN}^p_{k_i}(i)}{\operatorname{KNN}^q_{l_i}(i)}+ \frac{1}{|\trajectoryset^p|}
514 | \sum_{i=1}^{|\trajectoryset^p|} [\psi(l_i) - \psi(k_i)] +
515 | \log\frac{|\trajectoryset^q|}{|\trajectoryset^p|-1},
516 | \end{equation}
517 | $$
518 | where $\observationdim$ is the dimensionality of the trajectories, $|\trajectoryset^p|$ is the number of
519 | trajectories in the $\trajectoryset^{p}$ dataset, $|\trajectoryset^q|$ is the number of particles in the
520 | $\trajectoryset^{q}$ dataset, $\operatorname{KNN}^p_{k_i}(i)$ is the distance from trajectory $\trajectory_i
521 | \in \trajectoryset^{p}$ to its $k_i$-th nearest neighbor in $\trajectoryset^{q}$,
522 | $\operatorname{KNN}^q_{l_i}(i)$ is the distance from trajectory $\trajectory_i \in \trajectoryset^{p}$ to
523 | its $l_i$-th nearest neighbor in $\trajectoryset^{p} \backslash \trajectory_i$, and $\psi$ is the digamma
524 | function. Note that this approximation of KL divergence can also be applied to compare parameter
525 | distributions, as we show in the synthetic data experiment from subsection 5.1 of our main paper (cf. Figure
526 | 4 (a) and (b) from the main paper) where the ground-truth parameter distribution is known.
527 |
528 |
529 | Throughout this work, we set $k_i$ and $l_i$ to 3 as this reduces the bias in the approximation, but does
530 | not require a large amount of samples from the ground-truth distribution.
531 |
532 |
533 |
2 Experiments
534 |
We provide further technical details and results from the experiments we present in the main paper.
535 |
536 |
2.1 Differentiable Simulator
537 |
538 |
539 | Other than requiring a differentiable forward dynamics model which allows to simulate the system in its
540 | entirety following the Markov assumption, our proposed algorithm does not rely on a particular choice of
541 | dynamical system or simulator for which its parameters need to be estimated.
542 | For our experiments, we use the Tiny
544 | Differentiable Simulator [5] that implements end-to-end
545 | differentiable contact models and the Articulated Body Algorithm (ABA) [6] to
546 | compute the forward dynamics (FD) for articulated rigid-body mechanisms. Given joint positions $\jointpos$,
547 | velocities $\jointvel$, torques $\jointtorque$ in generalized coordinates, and external forces
548 | $\externalforce$, ABA calculates the joint accelerations $\jointacc$.
549 | We use semi-implicit Euler integration to advance the system dynamics in time for a time step $\Delta t$:
550 | $$
551 | \begin{align}
552 | \jointacc_{t+1} = \operatorname{ABA}(\jointpos_t, \jointvel_t, \jointtorque_t, \externalforce_t; \params),
553 | \qquad
554 | \jointvel_{t+1} = \jointvel_t + \jointacc_{t+1} \timestep, \qquad
555 | \jointpos_{t+1} = \jointpos_t + \jointvel_{t+1} \timestep.
556 | \end{align}
557 | $$
558 |
559 |
560 | The second-order ODE described by Equation 4 is lowered to a first-order system, with state
561 | $\statevec_t = \begin{bmatrix}\jointpos_t & \jointvel_t\end{bmatrix}$. Furthermore, we deal primarily with
562 | the discrete
563 | time-stepped dynamics function
564 | $\statevec_{t+1} = \fstep(\statevec_t, t, \params)$,
565 | assuming that $\timestep$ is constant. The function $\fsim(\params, \statevec_0)$ uses $\fstep$ iteratively
566 | to produce a trajectory of states $[\statevec]_{t=1}^T$ given an initial state $\statevec_0$ and the
567 | parameters $\params$. Many systems of practical interest for robotics are controlled by an external input.
568 | In our formulation for parameter estimation, we include controls as explicit dependencies on the time
569 | parameter $t$.
570 |
571 |
572 | For an articulated rigid body system, the parameters $\params$ may include (but are not limited to) the
573 | masses, inertial properties and geometrical properties of the bodies in the mechanism, as well as joint and
574 | contact friction coefficients. Given $\frac{\partial \fstep}{\partial\params}$ and
575 | $\frac{\partial\fstep}{\partial\statevec}$, gradients of simulation parameters with respect to the state
576 | trajectories can be computed directly through the chain rule, or via the adjoint sensitivity
577 | method [7].
578 |
579 |
580 |
581 |
2.2 Identify Real-world Double Pendulum
582 |
583 | We set up the double pendulum estimation experiment with the 11 parameters shown in
584 | Table 1 to be estimated. The state space consists of the two
585 | positions and
586 | velocities of both joints: $\statevec = \begin{bmatrix}\mathbf{q}_{0:1} &
587 | \mathbf{\dot{q}}_{0:1}\end{bmatrix}$. The dataset of
588 | trajectories contains image sequences (see time lapse of an excerpt from a trajectory in
589 | Figure 2) and annotated pixel coordinates of the three vertices in
590 | the double
591 | pendulum, from which we extracted joint positions and velocities (via finite differencing given the known
592 | recording frequency of 400Hz).
593 |
594 |
595 |
596 |
597 |
598 |
599 |
600 |
601 | | Link |
602 | Parameter |
603 | Minimum |
604 | Maximum |
605 |
606 |
607 |
608 |
609 | | Link 1 |
610 | Mass |
611 | 0.05 |
612 | $\kg$ |
613 | 0.5 |
614 | $\kg$ |
615 |
616 |
617 | | $I_{xx}$ |
618 | 0.002 |
619 | $\kg\cdot\meter^2$ |
620 | 1.0 |
621 | $\kg\cdot\meter^2$ |
622 |
623 |
624 | | COM $x$ |
625 | -0.2 |
626 | $\meter$ |
627 | 0.2 |
628 | $\meter$ |
629 |
630 |
631 | | COM $y$ |
632 | -0.2 |
633 | $\meter$ |
634 | 0.2 |
635 | $\meter$ |
636 |
637 |
638 | | Joint friction |
639 | 0.0 |
640 | |
641 | 0.5 |
642 | |
643 |
644 |
645 | | Link 2
646 | | Length |
647 | 0.08 |
648 | $\meter$ |
649 | 0.3 |
650 | $\meter$ |
651 |
652 |
653 | | Mass |
654 | 0.05 |
655 | $\kg$ |
656 | 0.5 |
657 | $\kg$ |
658 |
659 |
660 | | $I_{xx}$ |
661 | 0.002 |
662 | $\kg\cdot\meter^2$ |
663 | 1.0 |
664 | $\kg\cdot\meter^2$ |
665 |
666 |
667 | | COM $x$ |
668 | -0.2 |
669 | $\meter$ |
670 | 0.2 |
671 | $\meter$ |
672 |
673 |
674 | | COM $y$ |
675 | -0.2 |
676 | $\meter$ |
677 | 0.2 |
678 | $\meter$ |
679 |
680 |
681 | | Joint friction |
682 | 0.0 |
683 | |
684 | 0.5 |
685 | |
686 |
687 |
688 |
689 |
690 |
691 |
692 |
693 |
694 |
695 |
696 |
697 | Table 1. Parameters to be estimated. $I$ refers to the $3\times3$ inertia matrix, COM stands
698 | for center of mass.
699 |
700 |
701 |
702 |
Figure 2. Time lapse of a double pendulum trajectory from the IBM
703 | dataset [8].
704 |
705 |
706 |
707 | Since we know that all trajectories in this dataset stem from the same double
708 | pendulum [8], we only used a single reference trajectory as target trajectory
709 | $\trajectory\real$ during the estimation. We let each estimator run for 2000 iterations. For evaluation, we
710 | calculate the consistency metrics from Tab. 1 of the main paper over 10 held-out trajectories from a test
711 | dataset. For comparison, we visualize the trajectory density over simulations rolled out from the last 100
712 | Markov samples (or 100 particles in the case of particle-based approaches) in
713 | Figure 3. The ground-truth shown in these plots again stems from
714 | the unseen test
715 | dataset. This experiment further demonstrates the generalizability of simulation-based inference, which,
716 | when an adequate model has been implemented and its parameters identified, can predict outcomes under
717 | various novel conditions even though the training dataset consisted of only a single trajectory in this
718 | example.
719 |
720 |
721 |
722 |
723 |
724 |
725 |
726 |
727 | |
728 | Emcee |
729 |
730 | 
731 | |
732 |
733 |
734 | |
735 | CEM |
736 |
737 | 
738 | |
739 |
740 |
741 | |
742 | NUTS |
743 |
744 | 
745 | |
746 |
747 |
748 | |
749 | SGLD |
750 |
751 | 
752 | |
753 |
754 |
755 | |
756 | SVGD |
757 |
758 | 
759 | |
760 |
761 |
762 | |
763 | CSVGD |
764 |
765 | 
766 | |
767 |
768 |
769 |
770 | Figure 3. Kernel density estimation over trajectory roll-outs from the last estimated 100
771 | parameter guesses of each method, applied to the physical double pendulum dataset from
772 | Section 2.2. The ground-truth trajectory here stems from the test
773 | dataset of 10
774 | trajectories that were held out during training. The particle-based approaches (CEM, SVGD, CSVGD)
775 | use 100 particles.
776 |
777 |
778 |
779 |
780 |
781 |
2.3 Ablation Study on Multiple Shooting
782 |
783 |
784 | We evaluate the baseline estimation algorithms with our proposed multiple-shooting likelihood
785 | function
786 | (using
787 | 10 shooting windows) on the physical double pendulum dataset from before. To make the constrained
788 | optimization problem amenable to the MCMC samplers, we formulate the defect constraint through the
789 | following
790 | likelihood:
791 | $$
792 | \begin{align}
793 | \pdef(\statevec_t^s, \statevec_t) = \mathcal{N}(\statevec_t^s | \statevec_t,\sigma^2_{\text{def}})
794 | \qquad
795 | t\in [h, 2h, \dots],
796 | \end{align}
797 | $$
798 | where we tune $\sigma^2_{\text{def}}$ to a small value (on the order of $10^{-2}$) such that the
799 | defects are
800 | minimized during the estimation. As we describe in Sec. 4.4 from our main paper, the parameter space
801 | is
802 | augmented by the shooting variables $\statevec_t^s$.
803 |
804 |
805 | As shown in Table 2, the MCMC approaches Emcee and NUTS do not benefit
806 | meaningfully from
807 | the multiple-shooting approach. Emcee often yields unstable simulations from which we are not able
808 | to
809 | compute some of the metrics. The increased dimensionality of the parameter space appears to add a
810 | significant challenge to these methods, which are known to scale poorly to higher dimensions.
811 | Despite being configured to use a Gaussian mixture model of 3 kernels, the CEM posterior immediately
812 | collapses to a single point such that the KL divergence of simulated against real trajectories
813 | cannot be computed.
814 |
815 |
816 |
817 |
818 |
819 | |
820 | $d_{\text{KL}} (\trajectoryset\real
821 | \parallel
822 | \trajectoryset\simu)$ |
823 | $d_{\text{KL}} (\trajectoryset\simu
824 | \parallel
825 | \trajectoryset\real)$ |
826 | MMD |
827 |
828 |
829 | | Algorithm |
830 | SS |
831 | MS |
832 | SS |
833 | MS |
834 | SS |
835 | MS |
836 |
837 |
838 |
839 |
840 | | Emcee |
841 | 8542.2466 |
842 | 8950.4574 |
843 | 4060.6312 |
844 | N/A |
845 | 1.1365 |
846 | N/A |
847 |
848 |
849 | | CEM |
850 | 8911.1798 |
851 | 8860.5115 |
852 | 8549.5927 |
853 | N/A |
854 | 0.9687 |
855 | 0.5682 |
856 |
857 |
858 | | SGLD |
859 | 8788.0962 |
860 | 5863.2728 |
861 | 7876.0310 |
862 | 2187.2825 |
863 | 2.1220 |
864 | 0.0759 |
865 |
866 |
867 | | NUTS |
868 | 9196.7461 |
869 | 8785.5326 |
870 | 6432.2131 |
871 | 4935.8983 |
872 | 0.5371 |
873 | 1.1642 |
874 |
875 |
876 | | (C)SVGD |
877 | 8803.5683 |
878 | 5204.5336 |
879 | 10283.6659 |
880 | 2773.1751 |
881 | 0.7177 |
882 | 0.0366 |
883 |
884 |
885 |
886 |
887 | Table 2. Consistency metrics of the posterior distributions approximated from the physical
888 | double pendulum dataset (Section 2.2) by the different estimation algorithms
889 | using the
890 | single-shooting likelihood $p_{ss}(\trajectory\real | \params)$ (column "SS") and the
891 | multiple-shooting
892 | likelihood $p_{ms}(\trajectory\real | \params)$ (column "MS") with 10 shooting windows. Note that
893 | SVGD with multiple-shooting corresponds to CSVGD.
894 |
895 |
896 |
897 | We observe a significant improvement in estimation accuracy on SGLD, where the multiple-shooting
898 | approach
899 | allowed it to converge to closely matching trajectories, as shown in Figure 4. As
900 | with
901 | SVGD,
902 | the availability of gradients allows this method to scale better to the higher dimensional parameter
903 | space,
904 | while the smoothed likelihood landscape further helps the approach to find better fitting
905 | parameters.
906 |
907 |
908 |
909 |
910 |
911 |
912 |
913 | Figure 4. Kernel density estimation over trajectory roll-outs from the last estimated 100
914 | parameter guesses of SGLD with the multiple-shooting likelihood model (see
915 | Section 2.3), applied to the physical double pendulum
916 | dataset from
917 | Section 2.2. Similarly to SVGD, SGLD benefits significantly from the
918 | smoother
919 | likelihood function while being able to cope with the augmented parameter space thanks to its
920 | gradient-based approach.
921 |
922 |
923 |
924 |
925 |
926 |
2.4 Comparison to Likelihood-free Inference
927 |
928 | Our Bayesian inference approach leverages the simulator as part of the likelihood model to
929 | approximate posterior distributions over simulation parameters, which means the simulator is indispensable
930 | in our estimation process}. In the following, we compare our approach against the likelihood-free inference
931 | approach BayesSim [9] that leverages approximate Bayesian computation (ABC)
932 | which is
933 | the most popular family of algorithms within likelihood-free methods.
934 |
935 |
936 |
937 | Likelihood-free methods assume the simulator is a black box that can generate a trajectory given a parameter
938 | setting. Instead of querying the simulator to evaluate the likelihood (as in our approach), a conditional
939 | density $q(\params | \trajectoryset)$ is learned directly from a dataset of simulation parameters and their
940 | corresponding rolled-out trajectories via supervised learning to approximate the posterior. A common choice
941 | of model for such density is a mixture density network [10], which
942 | parameterizes a
943 | Gaussian mixture model. This is in contrast to our (C)SVGD algorithm which can approximate
944 | any shape of posterior by being a nonparametric inference algorithm.
945 |
946 |
947 |
948 |
949 | For our experiments we collect a dataset of 10,000 simulated trajectories of parameters randomly sampled
950 | from the prior distribution. We train the density network via the Adam optimizer with a learning rate of
951 | $10^{-3}$ for 3000 epochs, after which we observed no meaningful improvement to the calculated
952 | log-likelihood loss during training. In the following, we provide further details on the likelihood-free
953 | inference pipeline we consider, by describing the input data processing and the model used for approximating
954 | the posterior.
955 |
956 |
957 |
958 | The input to the learned density model has to be a sufficient statistic of the underlying data, while being
959 | low-dimensional in order to keep the learning problem computationally tractable. We consider the following
960 | four methods of processing the trajectories that are the input to the likelihood-free methods, as
961 | visualized for an example trajectory in Figure 5. Note that we configured
962 | the following
963 | input processing methods to generate a one-dimensional input vector that has a reasonable length to be
964 | computationally feasible to train on (given the 10,000 trajectories from the training dataset), while
965 | achieving acceptable performance which we validated through testing various settings.
966 |
967 |
968 |
969 | Downsampled: we down-sample the trajectory so that for the double pendulum experiment
970 | (Section 2.2) we use only every 20th state, for the Panda arm experiment
971 | (Section 2.5) only every 200-th state of the trajectory. Finally, the state
972 | dimensions per
973 | trajectory are concatenated to a one-dimensional vector.
974 |
975 |
976 | Difference: we adapt the input statistic from the original BayesSim approach in [9, Eq. (22)] where the differences of two consecutive states along the
978 | trajectory are used in
979 | concatenation with their mean and variance:
980 | $$
981 | \psi(\trajectory) = (\operatorname{downsample}(\tau), \mathbb{E}[\tau], \operatorname{Var}[\tau])
982 | \qquad\text{where}\qquad
983 | \tau = \{\statevec_t - \statevec_{t-1}\}_{t=1}^T
984 | $$
985 | As before, we down-sample these state differences and concatenate them to a vector.
986 |
987 |
988 | Summary: for each state dimension of the trajectory, we compute the following statistics typical for
989 | time series: mean, variance, cross correlation between state dimensions of the trajectory, as well as
990 | auto-correlations for each dimension at 5 different time delays: [5, 10, 20, 50, 100] time steps.
991 | These numbers are concatenated for all state dimensions to a one-dimensional vector per input
992 | trajectory.
993 |
994 |
995 | Signature: we compute the signature transform from the signatory package [11]
997 | over the input trajectory. Such so-called path signatures have been recently introduced to extract
998 | information about order and area, thereby preserving features inherent to nonlinear trajectories. We select
999 | a depth for the signature transform of 3 for the double pendulum experiment, and 2 for the Panda arm
1000 | experiment, to obtain feature vectors of comparable size to the aforementioned input techniques.
1001 |
1002 |
1003 |
1027 |
1028 | Figure 5. Exemplary visualization of the input processing methods for the likelihood-free baselines
1029 | from Section 2.4 applied to a trajectory from the double pendulum
1030 | experiment in
1031 | Section 2.2.
1032 |
1033 |
1034 |
2.4.2 Density Model
1035 |
1036 | As the density model for the learned posterior $q(\params | \trajectoryset)$, we select the following
1037 | commonly used representations.
1038 |
1039 |
1040 | Mixture density network (MDN): uses neural network features from a feed-forward neural network
1041 | using two hidden layers with 24 units each.
1042 |
1043 |
1044 | Mixture density random Fourier features (MDRFF): this density model uses Fourier features and
1045 | a kernel. We evaluate the MDRFF with the following common choices for the kernel:
1046 |
1047 | - Radial Basis Function (RBF) kernel
1048 | - Matérn kernel [12, Eq. (4.14)] with $\nu=\frac{5}{2}$
1049 |
1050 |
1051 |
2.4.3 Evaluation
1052 |
1053 | Note that instead of action generation, which is part of the proposed BayesSim
1054 | pipeline [9], we only focus on the inference of the posterior density over
1055 | simulation
1056 | parameters in order to compare such likelihood-free inference approach against our method.
1057 |
1058 |
1059 | To evaluate the metrics shown in Table 2 for each BayesSim
1060 | instantiation (input method and density model), we sample 100 parameter vectors from the learned
1061 | posterior $q(\params | \trajectoryset)$ and simulate them to obtain 100 trajectories which are compared
1062 | against the reference trajectory sets, as we did in the comparison for the other Bayesian inference methods
1063 | in Tab. 1 of the main paper.
1064 |
1065 |
1066 |
1067 |
1068 | |
1069 | |
1070 | Double Pendulum Experiment
1071 | |
1072 | Panda Arm Experiment |
1073 |
1074 |
1075 | | Input |
1076 | Model |
1077 | $d_{\text{KL}} (\trajectoryset\real \parallel
1078 | \trajectoryset\simu)$
1079 | |
1080 | $d_{\text{KL}} (\trajectoryset\simu \parallel \trajectoryset\real)$
1081 | |
1082 | MMD
1083 | |
1084 | $\log\pobs(\trajectoryset\real \parallel \trajectoryset\simu)$ |
1085 |
1086 |
1087 |
1088 |
1089 | | Downsampled |
1090 | MDN |
1091 | 8817.9222 |
1092 | 4050.4666 |
1093 | 0.6748 |
1094 | -17.4039 |
1095 |
1096 |
1097 | | Difference |
1098 | MDN |
1099 | 8919.2463 |
1100 | 4633.2637 |
1101 | 0.6285 |
1102 | -17.1646 |
1103 |
1104 |
1105 | | Summary |
1106 | MDN |
1107 | 9092.5575 |
1108 | 5093.8851 |
1109 | 0.5664 |
1110 | -18.3481 |
1111 |
1112 |
1113 | | Signature |
1114 | MDN |
1115 | 8985.8056 |
1116 | 4610.5438 |
1117 | 0.5807 |
1118 | -19.3432 |
1119 |
1120 |
1121 | | Downsampled |
1122 | MDRFF (RBF) |
1123 | 9027.9474 |
1124 | 5091.5283 |
1125 | 0.5593 |
1126 | -17.2335 |
1127 |
1128 |
1129 | | Difference |
1130 | MDRFF (RBF) |
1131 | 8936.3823 |
1132 | 4282.8599 |
1133 | 0.5988 |
1134 | -18.4892 |
1135 |
1136 |
1137 | | Summary |
1138 | MDRFF (RBF) |
1139 | 9063.1753 |
1140 | 4884.1398 |
1141 | 0.5672 |
1142 | -19.5430 |
1143 |
1144 |
1145 | | Signature |
1146 | MDRFF (RBF) |
1147 | 8980.9080 |
1148 | 4081.1160 |
1149 | 0.6016 |
1150 | -18.3458 |
1151 |
1152 |
1153 | | Downsampled |
1154 | MDRFF (Matérn) |
1155 | 8818.1830 |
1156 | 3794.9873 |
1157 | 0.6110 |
1158 | -17.6395 |
1159 |
1160 |
1161 | | Difference |
1162 | MDRFF (Matérn) |
1163 | 8859.2156 |
1164 | 4349.9971 |
1165 | 0.6176 |
1166 | -17.2752 |
1167 |
1168 |
1169 | | Summary |
1170 | MDRFF (Matérn) |
1171 | 8962.0501 |
1172 | 4241.4551 |
1173 | 0.5999 |
1174 | -19.6672 |
1175 |
1176 |
1177 | | Signature |
1178 | MDRFF (Matérn) |
1179 | 9036.9626 |
1180 | 4620.9517 |
1181 | 0.5715 |
1182 | -18.1652 |
1183 |
1184 |
1185 | | CSVGD |
1186 | 5204.5336 |
1187 | 2773.1751 |
1188 | 0.0366 |
1189 | -15.1671 |
1190 |
1191 |
1192 |
1193 |
1194 | Table 2.
1195 | Consistency metrics of the posterior distributions approximated by the different BayesSim instantiations,
1196 | where the input method and model name (including the kernel type for the MDRFF model) are given. Each metric
1197 | is calculated across simulated and real trajectories. Lower is better on all metrics except the
1198 | log-likelihood $\log\pobs(\trajectoryset\real \parallel \trajectoryset\simu)$ from the Panda arm experiment.
1199 | For comparison, in the last row, we reproduce the numbers from CSVGD shown in Tab. 1 of the main paper.
1200 |
1201 |
1202 |
1203 |
1204 |
1205 |
1206 | 
1208 | MDN - Downsampled
1209 |
1210 | |
1211 |
1212 | 
1214 | MDN - Difference
1215 | |
1216 |
1217 |
1218 |
1219 | 
1221 | MDN - Summary
1222 | |
1223 |
1224 | 
1226 | MDN - Signature
1227 | |
1228 |
1229 |
1230 |
1231 | 
1233 | MDRFF (RBF) - Downsampled
1234 | |
1235 |
1236 | 
1238 | MDRFF (RBF) - Difference
1239 | |
1240 |
1241 |
1242 |
1243 | 
1245 | MDRFF (RBF) - Summary
1246 | |
1247 |
1248 | 
1250 | MDRFF (RBF) - Signature
1251 | |
1252 |
1253 |
1254 |
1255 | 
1257 | MDRFF (Matérn) - Downsampled
1258 | |
1259 |
1260 | 
1262 | MDRFF (Matérn) - Difference
1263 | |
1264 |
1265 |
1266 |
1267 | 
1269 | MDRFF (Matérn) - Summary
1270 | |
1271 |
1272 | 
1274 | MDRFF (Matérn) - Signature
1275 | |
1276 |
1277 |
1278 |
1279 | Figure 6. Kernel density estimation over trajectory roll-outs from 100 parameter samples drawn from
1280 | the posterior of each BayesSim method (model name with kernel choice in bold font + input method, see
1281 | Section 2.4), applied to the physical double pendulum dataset from
1282 | Section 2.2. The ground-truth trajectory here stems from the test dataset of
1283 | 10 trajectories
1284 | that were held out during training.
1285 |
1286 |
1287 |
2.4.4 Discussion
1288 |
1289 |
1290 | The results from our experiments with the various likelihood-free approaches in
1291 | Table 2 indicate that, among the tested pipelines, the MDRFF model
1292 | with Matérn
1293 | kernel and downsampled trajectory input overall performed the strongest, followed by the MDN with
1294 | downsampled input. In comparison to the likelihood-based algorithms from Tab. 1 of the main paper, these
1295 | results are comparable on the double pendulum experiment. However, in comparison to CSVGD, the estimated
1296 | likelihood-free posteriors are significantly less accurate, which can also be clearly seen in the density
1297 | plots over the rolled out trajectories from such learned densities in
1298 | Figure 6. On the Panda arm experiment, the likelihood-free
1299 | methods are
1300 | outperformed by the likelihood-based algorithms (such as the Emcee sampler) more often on the likelihood of
1301 | the learned parameter densities. CSVGD again achieves a much more accurate posterior in this experiment than
1302 | any likelihood-free approach.
1303 |
1304 |
1305 |
Why do these likelihood-free methods perform so poorly on a seemingly simple double pendulum?
1306 |
1307 |
1308 | One would expect that this kind of dynamical system poses no greater challenge to BayesSim when it was shown
1309 | to identify a cartpole's link length and cart mass successfully [9]. To
1310 | investigate
1311 | this problem, we revisit the simplified double pendulum estimation experiment from Sec. 5.1 of our main
1312 | paper, where only the two link lengths need to be estimated from simulated trajectories. As before, we
1313 | create a dataset with 10,000 trajectories of 400 time steps based on the two simulation parameters sampled
1314 | from a uniform distribution ranging between 0.5m and 5m. While keeping all
1315 | parameters the same as in our previous double-pendulum experiment where eleven parameters had to be
1316 | inferred, all of the density models in combination with both the "difference" and "downsampled" input
1317 | statistic infer a highly accurate parameter distribution, as shown in
1318 | Figure 7 (a). The trajectories produced by sampling from the
1319 | BayesSim posterior
1320 | (Figure 7 (b)) also match the reference observations much
1321 | more closely than
1322 | any of the BayesSim models on the previous 11-parameter double
1323 | pendulum (Figure 6). These results suggest that BayesSim and
1324 | potentially
1325 | other likelihood-free method have problems in inferring higher dimensional parameter distributions. The
1326 | experiments in [9] demonstrated as many as four parameters being estimated
1327 | (for the
1328 | acrobot), while showing inference results for simulated systems only. While our double pendulum system from
1329 | Section 2.2 is basic in principle, the higher dimensional parameter space
1330 | (see parameters in
1331 | Table 1) and the fact that we need to fit against real-world data
1332 | makes it a
1333 | significantly harder problem for most state-of-the-art inference algorithms. CSVGD is able to achieve a
1334 | close fit thanks to the multiple-shooting segmentation of the trajectory which improves the convergence (see
1335 | more ablation results for multiple-shooting on this experiment in Section 2.3).
1337 |
1338 |
1339 |
1340 |
1341 |
1342 |
1343 |
(a) Posterior
1344 |
1345 |
1346 |
1348 |
(b) Trajectory density
1349 |
1350 |
1351 |
1352 |
1353 |
1354 |
1355 |
1356 | Figure 7. Results from BayesSim on the simplified double pendulum experiment where only the
1357 | two link lengths need to be inferred. (a) shows the approximated posterior distribution by the MDN
1358 | model and "downsampled" input statistics. The diagonal plots show the marginal parameter
1359 | distributions, the bottom-left heatmap and the top-right contour plot show the 2D posterior where
1360 | the ground-truth parameters at (1.5m, 2m) are indicated by a red star. The
1361 | black dots in the top-right plot are 100 parameters sampled from the posterior which are rolled out
1362 | to generate trajectories for the trajectory density plot in (b). (b) shows a kernel density
1363 | estimation over these 100 trajectories for the four state dimensions
1364 | $(\jointpos_{[0:1]},\jointvel_{[0:1]})$ of the double pendulum.
1365 |
1366 |
1367 |
1368 |
1369 |
1370 |
2.5 Identify Inertia of an Articulated Rigid Object
1371 |
1372 | The state space consists of the positions and velocities of the seven degrees of freedom of the robot arm
1373 | and the two degrees of freedom in the universal joint, resulting in a 20-dimensional state vector
1374 | $\statevec=\begin{bmatrix}\mathbf{q}_{0:8} & \mathbf{\dot{q}}_{0:8} & \mathbf{q}^d_{0:6} &
1375 | \mathbf{\dot{q}}^d_{0:6}\end{bmatrix}$
1376 | consisting of nine joint positions and velocities, plus the PD control position and velocity targets,
1377 | $\mathbf{q}^d$ and $\mathbf{\dot{q}}^d$, for the actuated joints of the robot arm. We control the arm using
1378 | the MoveIt! motion planning framework [13] by moving joints 6 and 7
1379 | to
1380 | predefined joint-space offsets of $0.1$ and $-0.1$ radians, in sequence. We use the default Iterative
1381 | Parabolic Time Parameterization algorithm with a velocity scaling factor of $0.1$. We track the motion of
1382 | the acrylic box via a Vicon motion capture system and derive four Cartesian coordinates as observation
1383 | $\observationvec=\begin{bmatrix}\mathbf{p}_{o} & \mathbf{p}_{x} & \mathbf{p}_{y} &
1384 | \mathbf{p}_{z}\end{bmatrix}$ to represent the
1385 | frame of the box (shown in Figure 8): a point of origin located at the
1386 | center of the
1387 | upper lid of the box, and three points located 1m away from the origin into the x, y, and z
1388 | direction (transformed by the reference frame of the box). We chose this state representation to ease the
1389 | computation of the likelihood, since we only need to compute differences between 3D points instead of
1390 | computing the distances over 3D rotations which requires special treatment [14].
1392 |
1393 |
1394 |
1395 |
1396 |
1397 |
1398 | Figure 8. Rendering of the simulation for the underactuated mechanism from
1399 | Section 2.5, where the four reference points for the origin, unit x, y,
1400 | and z vectors
1401 | are shown. The trace of the simulated trajectory is visualized by the solid lines, the ground-truth
1402 | trajectories of the markers are shown as dotted lines.
1403 |
1404 |
1405 |
1406 |
1407 |
1408 | We first identify the simulation parameters pertaining to the inertial properties of the box and the
1409 | friction parameters of the universal joint. As shown in Table 3
1410 | (a), the symmetric
1411 | inertia matrix of the box is fully determined by the first six parameters, followed by the 3D center of
1412 | mass. We have measured the mass to be 920g, so we do not need to estimate it. We simulate the
1413 | universal joint with velocity-dependent damping, with possibly different friction coefficients for both
1414 | degrees of freedom. The simulation parameters yielding the most accurate fit to a ground-truth trajectory
1415 | from the physical robot shaking an empty box is shown in Figure 9.
1416 | We were able to
1417 | find such parameters via SVGD, CSVGD and Emcee (shown is a parameter configuration from the particle
1418 | distribution estimated by CSVGD with the highest likelihood).
1419 |
1420 |
1421 |
1422 | While the simulated trajectory matches the real data significantly better after the inertial parameters of
1423 | the empty box have been identified (Figure 9 (b)) than before
1424 | (Figure (a)), a reality gap remains. We believe this to be a
1425 | result from a
1426 | slight modeling error that the rigid body simulator cannot capture, e.g. the top of the box where the
1427 | universal joint is attached bends slightly while the box is moving, and there may be slight geometric
1428 | offsets between the real system and the model of it we use in the simulator. The latter parameters could
1429 | have been further identified with our approach, nonetheless the simulation given the identified parameters
1430 | is sufficient to be used in the next phase of the inference experiment.
1431 |
1432 |
1433 |
1434 |
1435 |
1436 |
(a) Before identification of empty box
1437 |
1438 |
1439 |
1440 |
(b) After identification of empty box
1441 |
1442 |
1443 |
1444 | Figure 9. Trajectories from the Panda robot arm shaking an empty box. Visualized are the
1445 | simulated (red) and real (black) observations before (a) and after (b) the inertial parameters of
1446 | the empty box and the friction from the universal joint (Table 3
1447 | (a)) have been
1448 | identified. The columns correspond to the four reference points in the frame of the box (see a rendering of
1449 | them in Figure 8), the rows show the $x$, $y$, and $z$ axes of these
1450 | reference points in
1451 | meters. The horizontal axes show the time step.
1452 |
1453 |
1454 |
1455 | Given the parameters found in the first phase, we now investigate how well the various approaches can cope
1456 | with dependent variables. By fixing two 500g to the bottom of the acrylic box, the 2D locations
1457 | of such weights need to be inferred. Naturally, such assignment is symmetric, i.e. weight 1 and 2 can swap
1458 | locations without affecting the dynamics. What would significantly alter the dynamics, however, is an
1459 | unbalanced configuration of the weights which would cause the box to tilt.
1460 |
1461 |
1462 |
1463 |
1464 |
1465 |
1466 |
1467 |
1468 | |
1469 | Parameter |
1470 | Minimum |
1471 | Maximum |
1472 |
1473 |
1474 |
1475 |
1476 | | Box |
1477 | $I_{xx}$ |
1478 | 0.05 |
1479 | $\kg\cdot\meter^2$ |
1480 | 0.1 |
1481 | $\kg\cdot\meter^2$ |
1482 |
1483 |
1484 | | $I_{yy}$ |
1485 | 0.05 |
1486 | $\kg\cdot\meter^2$ |
1487 | 0.1 |
1488 | $\kg\cdot\meter^2$ |
1489 |
1490 |
1491 | | $I_{zz}$ |
1492 | 0.05 |
1493 | $\kg\cdot\meter^2$ |
1494 | 0.1 |
1495 | $\kg\cdot\meter^2$ |
1496 |
1497 |
1498 | | $I_{xy}$ |
1499 | -0.01 |
1500 | $\kg\cdot\meter^2$ |
1501 | 0.01 |
1502 | $\kg\cdot\meter^2$ |
1503 |
1504 |
1505 | | $I_{xz}$ |
1506 | -0.01 |
1507 | $\kg\cdot\meter^2$ |
1508 | 0.01 |
1509 | $\kg\cdot\meter^2$ |
1510 |
1511 |
1512 | | $I_{yz}$ |
1513 | -0.01 |
1514 | $\kg\cdot\meter^2$ |
1515 | 0.01 |
1516 | $\kg\cdot\meter^2$ |
1517 |
1518 |
1519 | | COM $x$ |
1520 | -0.005 |
1521 | $\meter$ |
1522 | 0.005 |
1523 | $\meter$ |
1524 |
1525 |
1526 | | COM $y$ |
1527 | -0.005 |
1528 | $\meter$ |
1529 | 0.005 |
1530 | $\meter$ |
1531 |
1532 |
1533 | | COM $z$ |
1534 | 0.1 |
1535 | $\meter$ |
1536 | 0.4 |
1537 | $\meter$ |
1538 |
1539 |
1540 |
1541 | | U-Joint |
1542 | Friction DOF 1 |
1543 | 0.0 |
1544 | |
1545 | 0.15 |
1546 | |
1547 |
1548 |
1549 | | Friction DOF 2 |
1550 | 0.0 |
1551 | |
1552 | 0.15 |
1553 | |
1554 |
1555 |
1556 |
1557 |
(a) Phase I
1558 |
1559 |
1560 |
1561 |
1562 |
1563 |
1564 |
1565 | |
1566 | Parameter |
1567 | Minimum |
1568 | Maximum |
1569 |
1570 |
1571 |
1572 |
1573 | | Weight 1 |
1574 | Position $x$ |
1575 | -0.14 |
1576 | $\meter$ |
1577 | 0.14 |
1578 | $\meter$ |
1579 |
1580 |
1581 | | Position $y$ |
1582 | -0.08 |
1583 | $\meter$ |
1584 | 0.08 |
1585 | $\meter$ |
1586 |
1587 |
1588 | | Weight 2 |
1589 | Position $x$ |
1590 | -0.14 |
1591 | $\meter$ |
1592 | 0.14 |
1593 | $\meter$ |
1594 |
1595 |
1596 | | Position $y$ |
1597 | -0.08 |
1598 | $\meter$ |
1599 | 0.08 |
1600 | $\meter$ |
1601 |
1602 |
1603 |
1604 |
(b) Phase II
1605 |
1606 |
1607 |
1608 |
1609 |
1610 | Table 3. Parameters to be estimated and their ranges for the two estimation phases of the
1611 | underactuated mechanism experiment from Section 2.5.
1612 |
1613 |
1614 |
1615 | We use 50 particles and run each estimation algorithm for 500 iterations. For each baseline method, we
1616 | carefully tuned the hyper parameters to facilitate a fair comparison. Such tuning included selecting an
1617 | appropriate measurement noise variance, which, as we observed on Emcee and SGLD in particular, had a
1618 | significant influence on the exploration behavior of these algorithms. With a larger observation noise
1619 | variance the resulting posterior distribution became wider, however we were unable to attain such behavior
1620 | with the NUTS estimator whose iterates quickly collapsed to a single point at the center of the box (see
1621 | Figure 10 (d)). Similarly, CEM immediately became stuck in the
1622 | suboptimal
1623 | configuration shown in Figure 10 (b). Nonetheless, after 500
1624 | iterations, all methods
1625 | predicted weight positions that were aligned opposite to one another to balance the box.
1626 |
1627 |
1628 |
1629 | As can be seen in Figure 10 (e), SVGD achieves a fairly
1630 | predictive posterior
1631 | approximation, with many particles aligned close to the true vertical position at $y=0$. With the
1632 | introduction of the multiple-shooting constraints, Constrained SVGD (CSVGD) converges significantly faster
1633 | to a posterior distribution that accurately captures the true locations of the box, while retaining the
1634 | exploration performance of SVGD that spreads out the particles over multiple modes, as shown in
1635 | Figure 10 (f).
1636 |
1637 |
1638 |
1639 |
1640 |
1641 |
(a) Emcee
1642 |
1643 |
1644 |
1645 |
(b) CEM
1646 |
1647 |
1648 |
1649 |
(c) SGLD
1650 |
1651 |
1652 |
1653 |
(d) NUTS
1654 |
1655 |
1656 |
1657 |
(e) SVGD
1658 |
1659 |
1660 |
1661 |
(f) CSVGD
1662 |
1663 |
1664 |
1665 | Figure 10. Posterior plots over the 2D weight locations approximated by the estimation algorithms
1666 | applied to the underactuated mechanism experiment from Section 2.5. Blue shades
1667 | indicate a
1668 | Gaussian kernel density estimation computed over the inferred parameter samples. Since it is an
1669 | unbounded kernel density estimate, the blue shades cross the parameter boundaries in certain areas (e.g. for
1670 | CSVGD), while in reality none of the estimated particles violate the parameter limits.
1671 |
1672 |
1673 |
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