nips nips2008 nips2008-125 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Duy Nguyen-tuong, Jan R. Peters, Matthias Seeger
Abstract: Learning in real-time applications, e.g., online approximation of the inverse dynamics model for model-based robot control, requires fast online regression techniques. Inspired by local learning, we propose a method to speed up standard Gaussian process regression (GPR) with local GP models (LGP). The training data is partitioned in local regions, for each an individual GP model is trained. The prediction for a query point is performed by weighted estimation using nearby local models. Unlike other GP approximations, such as mixtures of experts, we use a distance based measure for partitioning of the data and weighted prediction. The proposed method achieves online learning and prediction in real-time. Comparisons with other non-parametric regression methods show that LGP has higher accuracy than LWPR and close to the performance of standard GPR and ν-SVR. 1
Reference: text
sentIndex sentText sentNum sentScore
1 , online approximation of the inverse dynamics model for model-based robot control, requires fast online regression techniques. [sent-7, score-0.638]
2 Inspired by local learning, we propose a method to speed up standard Gaussian process regression (GPR) with local GP models (LGP). [sent-8, score-0.421]
3 The training data is partitioned in local regions, for each an individual GP model is trained. [sent-9, score-0.234]
4 The prediction for a query point is performed by weighted estimation using nearby local models. [sent-10, score-0.39]
5 The proposed method achieves online learning and prediction in real-time. [sent-12, score-0.226]
6 Especially in robot tracking control, only a well-estimated inverse dynamics model can allow both high accuracy and compliant control. [sent-15, score-0.437]
7 For most real-time applications, online model learning poses a difficult regression problem due to three constraints, i. [sent-17, score-0.219]
8 Here, the true function is approximated with local linear functions covering the relevant state-space and online learning became computationally feasible due to low computational demands of the local projection regression which can be performed in real-time. [sent-27, score-0.554]
9 , online learning of inverse dynamics model for model-based 1 robot control. [sent-35, score-0.424]
10 The computational cost is then significantly reduced due to much smaller number of training examples within a local model. [sent-42, score-0.244]
11 The ME performance depends largely on the way of partitioning the training data and the choice of an optimal number of local models for a particular data set [4]. [sent-43, score-0.329]
12 , LWPR and GPR, attempting to get as close as possible to the speed of local learning while having a comparable accuracy to Gaussian process regression. [sent-46, score-0.216]
13 This results in an approach inspired by [6, 8] using many local GPs in order to obtain a significant reduction of the computational cost during both prediction and learning step allowing the application to online learning. [sent-47, score-0.399]
14 For partitioning the training data, we use a distance based measure, where the corresponding hyperparameters are optimized by maximizing the marginal likelihood. [sent-48, score-0.175]
15 Subsequently, we describe our local Gaussian process models (LGP) approach in Section 3 and discuss how it inherits the advantages of both GPR and LWPR. [sent-50, score-0.205]
16 , LWPR [8], standard GPR [1], sparse online Gaussian process regression (OGP) [5] and ν-support vector regression (ν-SVR) [11], respectively. [sent-53, score-0.272]
17 Finally, our LGP method is evaluated for an online learning of the inverse dynamics models of real robots for accurate tracking control in Section 5. [sent-54, score-0.494]
18 Here, the online learning is demonstrated by rank-one update of the local GP models [9]. [sent-55, score-0.347]
19 The tracking task is performed in real-time using model-based control [10]. [sent-56, score-0.179]
20 To our best knowledge, it is the first time that GPR is successfully used for high-speed online model learning in real time control on a physical robot. [sent-57, score-0.228]
21 We present the results on a version of the Barrett WAM showing that with the online learned model using LGP the tracking accuracy is superior compared to state-of-the art model-based methods [10] while remaining fully compliant. [sent-58, score-0.328]
22 for k = 1 to M do Compute distance to the k-th local model: wk = exp(−0. [sent-69, score-0.307]
23 5(x − ck )T W(x − ck )) Compute local mean using the k-th local model: yk = kT αk ¯ k end for Compute weighted prediction using M local models: M M y = k=1 wk yk / k=1 wk . [sent-70, score-1.149]
24 for k = 1 to number of local models do Compute distance to the k-th local model: wk = exp(−0. [sent-72, score-0.488]
25 (a) SARCOS arm Algorithm 1: Partitioning of training data and model learning. [sent-75, score-0.168]
26 Reducing this computational cost, we cluster the training data in local regions and, subsequently, train the corresponding GP models on these local clusters. [sent-78, score-0.391]
27 The mean prediction for a query point is then made by weighted prediction using the nearby local models in the neighborhood. [sent-79, score-0.491]
28 , allocation of new input points and learning of corresponding local models, (ii) prediction for a query point. [sent-82, score-0.384]
29 1 Partitioning and Training of Local Models Clustering input data is efficiently performed by considering a distance measure of the input point x to the centers of all local models. [sent-84, score-0.273]
30 The distance measure wk is given by the kernel used to learn the local GP models, e. [sent-85, score-0.328]
31 , Gaussian kernel 1 T wk = exp − (x − ck ) W (x − ck ) , (4) 2 where ck denotes the center of the k-th local model and W a diagonal matrix represented the kernel width. [sent-87, score-0.669]
32 It should be noted, that we use the same kernel width for computing wk as well as for training of all local GP models as given in Section 2. [sent-88, score-0.392]
33 During the localization process, a new model with center ck+1 is created, if all distance measures wk fall below a limit value wgen . [sent-91, score-0.32]
34 Thus, the number of local models is allowed to increase as the trajectories become more complex. [sent-93, score-0.181]
35 Otherwise, if a new point is assigned to a particular k-th model, the center ck is updated as mean of corresponding local 3 data points. [sent-94, score-0.308]
36 With the new assigned input point, the inverse covariance matrix of the corresponding local model can be updated. [sent-95, score-0.279]
37 The main computational cost of this algorithm is O(N 3 ) for inverting the local covariance matrix, where N presents the number of data points in a local model. [sent-97, score-0.428]
38 Furthermore, we can control the complexity by limiting the number of data points in a local model. [sent-98, score-0.268]
39 Since the number of local data points increases continuously over time, we can adhere to comply with this limit by deleting old data point as new ones are included. [sent-99, score-0.284]
40 The cost for inverting the local covariance matrix can be further reduced, as we need only to update the full inverse matrix once it is computed. [sent-101, score-0.313]
41 2 Prediction using Local Models The prediction for a mean value y is performed using weighted averaging over M local predicˆ tions yk for a query point x [8]. [sent-104, score-0.46]
42 The weighted prediction y is then given by y = E{¯k |x} = ¯ ˆ ˆ y M yk p(k|x). [sent-105, score-0.18]
43 According to the Bayesian theorem, the probability of the model k given x can be ¯ k=1 M M expressed as p(k|x) = p(k, x)/ k=1 p(k, x) = wk / k=1 wk . [sent-106, score-0.302]
44 (5) The probability p(k|x) can be interpreted as a normalized distance of the query point x to the local model k where the measure metric wk is used as given in Equation (4). [sent-108, score-0.447]
45 Thus, each local prediction yk , determined using Equation (3), is additionally weighted by the distance wk between ¯ the corresponding center ck and the query point x. [sent-109, score-0.728]
46 The search for M local models can be quickly done by evaluating the distances between the query point x and all model centers ck . [sent-110, score-0.42]
47 4 Learning Inverse Dynamics We have evaluated our algorithm using high-dimensional robot data taken from real robots, e. [sent-112, score-0.152]
48 , the 7 degree-of-freedom (DoF) anthropomorphic SARCOS master arm and 7-DoF Barrett whole arm manipulator shown in Figure 1, as well as a physically realistic SL simulation [12]. [sent-114, score-0.2]
49 , LWPR, ν-SVR, OGP and standard GPR in the context of approximating inverse robot dynamics. [sent-117, score-0.201]
50 For the considered 7 degrees of freedom robot arms, we, thus, have data with 21 input dimensions (for each joint, we have an angle, a velocity and an acceleration) and 7 targets (a torque for each joint). [sent-122, score-0.285]
51 We learn the robot dynamics model in this 21-dim space for each DoF separately employing LWPR, ν-SVR, GPR, OGP and LGP, respectively. [sent-123, score-0.215]
52 Partitioning of the training examples for LGP can be performed either in the same input space (where the model is learned) or in another space which has to be physically consistent with the approximated function. [sent-124, score-0.165]
53 Thus, the partitioning of training data is performed in a 7-dim space (7 joint angles). [sent-126, score-0.172]
54 After determining wk for all k local models in the partitioning space, the input point will be assigned to the nearest local model, i. [sent-127, score-0.566]
55 , the local model with the maximal value of distance measure wk . [sent-129, score-0.331]
56 The error is computed after prediction on the test sets with simulated data from SL Sarcos-model, real robot data from Barrett and SARCOS master arm, respectively. [sent-145, score-0.302]
57 , the simulated SARCOS arm in (a), the real SARCOS arm in (b) and the Barrett arm in (c). [sent-151, score-0.225]
58 During the prediction on the test set using LGP, we take the most activated local models, i. [sent-153, score-0.225]
59 If wgen is too GPR small, a lot of local models will be generated LGP with small number of training points. [sent-157, score-0.312]
60 It turns 2 out that these small local models do not perform well in generalization for unknown data. [sent-158, score-0.181]
61 1 If wgen is large, the local models become also large which increase the computational complexity. [sent-159, score-0.261]
62 Here, the training data are clustered in about 30 local regions ensuring that each local model has a sufficient amount of data points 0 5000 10000 15000 for high accuracy (in practice, roughly a hunNr. [sent-160, score-0.469]
63 of Training Points Figure 3: Average time in millisecond needed for dred data points for each local model suffice) prediction of 1 query point. [sent-161, score-0.406]
64 This small number of training inputs methods such as standard GPR and ν-SVR, local enables a fast training for each local model, i. [sent-167, score-0.403]
65 05 7 6 5 4 3 Considering the approximation error on the test set shown in Figure 2(a-c), it can be seen that LGP generalizes well using only few local models for prediction. [sent-176, score-0.202]
66 The operation of mean-prediction has then the order of O(n) for standard GPR (similarly, for ν-SVR) and O(N M ) for LGP, where n denotes the total number of training points, M number of local models and N number of data points in a local model. [sent-180, score-0.438]
67 , matrix inversion), the prediction time is also reduced significantly compared to GPR and ν-SVR due to the fact that only a small amount of local models in the vicinity of the query point are needed during prediction for LGP. [sent-184, score-0.507]
68 Thus, the prediction time can be controlled by the number of local models. [sent-185, score-0.225]
69 A large number of local models may provide a smooth prediction but on the other hand increases the time complexity. [sent-186, score-0.265]
70 Subsequently, using the trained models we compute the average time needed to make a prediction for a query point for all 7 DoF. [sent-189, score-0.24]
71 For LGP, we take a limited number of local models in the vicinity for prediction, e. [sent-190, score-0.203]
72 Since our control system requires a minimal prediction rate at 100 Hz (10 ms) in order to ensure system stability, data sets with more than 15000 points are not applicable for standard GPR or ν-SVR due to high computation demands for prediction. [sent-193, score-0.26]
73 As approach to deleting and inserting data points, we can use the information gain of the corresponding local model as a principled measure. [sent-198, score-0.219]
74 , more than 5000 training examples), LGP reduces the prediction cost considerably while keeping a good learning performance. [sent-202, score-0.167]
75 5 qd ˙ qd ¨ qd Application in Model-based Robot Control Local GP u + Robot + + Kp ˙ q q Kv + − In this section, first, we use the inverse dynamics models learned in Section 4. [sent-203, score-0.473]
76 1 for a modelbased tracking control task [10] in the setting shown in Figure 4. [sent-204, score-0.156]
77 Here, the learned model of the robot is applied for an online prediction of the feedforward torques uFF given the desired trajectory [qd , qd , qd ]. [sent-205, score-0.71]
78 Subsequently, ˙ ¨ the model approximated by LGP is used for an online learning performance. [sent-206, score-0.19]
79 Demonstrating the online learning, the local GP models are adapted in real-time using rank-one update. [sent-207, score-0.323]
80 As shown in Figure 4, the controller command u consists of the feedforward part uFF and the feedback part uFB = Kp e + Kv e, where e = ˙ qd − q denotes the tracking error and Kp , Kv Figure 4: Schematic showing model-based robot position-gain and velocity-gain, respectively. [sent-208, score-0.37]
81 The learned dynamics model can be upDuring the control experiment we set the gains dated online using LGP. [sent-210, score-0.324]
82 As a result, the learned model has a stronger effect on computing the predicted torque uFF and, hence, a better learning performance of each method results in a lower tracking error. [sent-212, score-0.205]
83 + − For comparison with the learned models, we also compute the feedforward torque using rigid-body (RB) formulation which is a common approach in robot control [10]. [sent-213, score-0.314]
84 01 0 1 2 3 4 5 6 Degree of Freedom 0 7 (a) Tracking Error on Barrett without online learning 1 2 3 4 5 6 Degree of Freedom 7 (b) Tracking Error after LGP online learning on Barrett Figure 5: (a) Tracking error as RMSE on test trajectory for each DoF with Barrett WAM. [sent-227, score-0.339]
85 The model uncertainty is reduced with online learning using LGP. [sent-229, score-0.186]
86 With online learning, LGP is able to outperform offline learned models using standard GPR for test trajectories. [sent-230, score-0.221]
87 As desired trajectory, we generate a test trajectory which is similar to the one used for learning the inverse dynamics models in Section 4. [sent-232, score-0.198]
88 Figure 5 (a) shows the tracking errors on test trajectory for 7 DoFs, where the error is computed as root mean squared error (RMSE). [sent-234, score-0.17]
89 Since it is not possible to learn the complete state space using a single data set, online learning is necessary. [sent-244, score-0.16]
90 1 Online Learning of Inverse Dynamics Models with LGP The ability of online adaptation of the learned inverse dynamics models with LGP is shown by the rank-one update of the local models which has a complexity of O(n2 ) [9]. [sent-246, score-0.55]
91 Since the number of training examples in each local model is limited (500 points in average), the update procedure is fast enough for real-time application. [sent-247, score-0.306]
92 For online learning the models are updated as shown in Figure 4. [sent-248, score-0.182]
93 For doing so, we regularly sample the joint torques u and the corresponding robot trajectories [q, q, q] online. [sent-249, score-0.195]
94 For the time being, as a new point is inserted we randomly delete another data ˙ ¨ point from the local model if the maximal number of data point is reached. [sent-250, score-0.273]
95 Figure 5 (b) shows the tracking error after online learning with LGP. [sent-252, score-0.257]
96 It can be seen that the errors for each DoF are significantly reduced with online LGP compared to the ones with offline learned models. [sent-253, score-0.201]
97 6 Conclusion We combine with LGP the fast computation of local regression with more accurate regression methods while having little tuning efforts. [sent-255, score-0.285]
98 The reducing cost allows LGP for model online learning which is necessary in oder to generalize the model for all trajectories. [sent-257, score-0.222]
99 Model-based tracking control using online learned model achieves superior control performance compared to the state-of-the-art method as well as offline learned model for unknown trajectories. [sent-258, score-0.486]
100 Seeger, “Computed torque control with nonparametric regression models,” Proceedings of the 2008 American Control Conference (ACC 2008), 2008. [sent-335, score-0.163]
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