Many real-world systems can be described by mathematical models that are human-comprehensible, easy to analyze and help explain the system's behavior. Symbolic regression is a method that can automatically generate such models from data. Historically, symbolic regression has been predominantly realized by genetic programming, a method that evolves populations of candidate solutions that are subsequently modified by genetic operators crossover and mutation. However, this approach suffers from several deficiencies: it does not scale well with the number of variables and samples in the training data-models tend to grow in size and complexity without an adequate accuracy gain, and it is hard to fine-tune the model coefficients using just genetic operators. Recently, neural networks have been applied to learn the whole analytic model, i.e., its structure and the coefficients, using gradient-based optimization algorithms. This paper proposes a novel neural network-based symbolic regression method that constructs physically plausible models based on even very small training data sets and prior knowledge about the system. The method employs an adaptive weighting scheme to effectively deal with multiple loss function terms and an epoch-wise learning process to reduce the chance of getting stuck in poor local optima. Furthermore, we propose a parameter-free method for choosing the model with the best interpolation and extrapolation performance out of all the models generated throughout the whole learning process. We experimentally evaluate the approach on four test systems: the TurtleBot 2 mobile robot, the magnetic manipulation system, the equivalent resistance of two resistors in parallel, and the longitudinal force of the anti-lock braking system. The results clearly show the potential of the method to find parsimonious models that comply with the prior knowledge provided. @en

Novel view synthesis is a long-standing problem. In this work, we consider a variant of the problem where we are given only a few context views sparsely covering a scene or an object. The goal is to predict novel viewpoints in the scene, which requires learning priors. The current state of the art is based on Neural Radiance Field (NeRF), and while achieving impressive results, the methods suffer from long training times as they require evaluating millions of 3D point samples via a neural network for each image. We propose a 2D-only method that maps multiple context views and a query pose to a new image in a single pass of a neural network. Our model uses a two-stage architecture consisting of a codebook and a transformer model. The codebook is used to embed individual images into a smaller latent space, and the transformer solves the view synthesis task in this more compact space. To train our model efficiently, we introduce a novel branching attention mechanism that allows us to use the same model not only for neural rendering but also for camera pose estimation. Experimental results on real-world scenes show that our approach is competitive compared to NeRF-based methods while not reasoning explicitly in 3D, and it is faster to train.@en

Many real-world systems can be naturally described by mathematical formulas. The task of automatically constructing formulas to fit observed data is called symbolic regression. Evolutionary methods such as genetic programming have been commonly used to solve symbolic regression tasks, but they have significant drawbacks, such as high computational complexity. Recently, neural networks have been applied to symbolic regression, among which the transformer-based methods seem to be most promising. After training a transformer on a large number of formulas, the actual inference, i.e., finding a formula for new, unseen data, is very fast (in the order of seconds). This is considerably faster than state-of-the-art evolutionary methods. The main drawback of transformers is that they generate formulas without numerical constants, which have to be optimized separately, yielding suboptimal results. We propose a transformer-based approach called SymFormer, which predicts the formula by outputting the symbols and the constants simultaneously. This helps to generate formulas that fit the data more accurately. In addition, the constants provided by SymFormer serve as a good starting point for subsequent tuning via gradient descent to further improve the model accuracy. We show on several benchmarks that SymFormer outperforms state-of-the-art methods while having faster inference.@en

Many real-world systems can be naturally described by mathematical formulas. The task of automatically constructing formulas to fit observed data is called symbolic regression. Evolutionary methods such as genetic programming have been commonly used to solve symbolic regression tasks, but they have significant drawbacks, such as high computational complexity. Recently, neural networks have been applied to symbolic regression, among which the transformer-based methods seem to be most promising. After training a transformer on a large number of formulas, the actual inference, i.e., finding a formula for new, unseen data, is very fast (in the order of seconds). This is considerably faster than state-of-the-art evolutionary methods. The main drawback of transformers is that they generate formulas without numerical constants, which have to be optimized separately, yielding suboptimal results. We propose a transformer-based approach called SymFormer, which predicts the formula by outputting the symbols and the constants simultaneously. This helps to generate formulas that fit the data more accurately. In addition, the constants provided by SymFormer serve as a good starting point for subsequent tuning via gradient descent to further improve the model accuracy. We show on several benchmarks that SymFormer outperforms state-of-the-art methods while having faster inference.@en

Reinforcement learning algorithms can solve dynamic decision-making and optimal control problems. With continuous-valued state and input variables, reinforcement learning algorithms must rely on function approximators to represent the value function and policy mappings. Commonly used numerical approximators, such as neural networks or basis function expansions, have two main drawbacks: They are black-box models offering little insight into the mappings learned, and they require extensive trial and error tuning of their hyper-parameters. In this paper, we propose a new approach to constructing smooth value functions in the form of analytic expressions by using symbolic regression. We introduce three off-line methods for finding value functions based on a state-transition model: Symbolic value iteration, symbolic policy iteration, and a direct solution of the Bellman equation. The methods are illustrated on four nonlinear control problems: Velocity control under friction, one-link and two-link pendulum swing-up, and magnetic manipulation. The results show that the value functions yield well-performing policies and are compact, mathematically tractable, and easy to plug into other algorithms. This makes them potentially suitable for further analysis of the closed-loop system. A comparison with an alternative approach using neural networks shows that our method outperforms the neural network-based one. @en

Continual model learning for nonlinear dynamic systems, such as autonomous robots, presents several challenges. First, it tends to be computationally expensive as the amount of data collected by the robot quickly grows in time. Second, the model accuracy is impaired when data from repetitive motions prevail in the training set and outweigh scarcer samples that also capture interesting properties of the system. It is not known in advance which samples will be useful for model learning. Therefore, effective methods need to be employed to select informative training samples from the continuous data stream collected by the robot. Existing literature does not give any guidelines as to which of the available sample-selection methods are suitable for such a task. In this paper, we compare five sample-selection methods, including a novel method using the model prediction error. We integrate these methods into a model learning framework based on symbolic regression, which allows for learning accurate models in the form of analytic equations. Unlike the currently popular data-hungry deep learning methods, symbolic regression is able to build models even from very small training data sets. We demonstrate the approach on two real robots: the TurtleBot mobile robot and the Parrot Bebop drone. The results show that an accurate model can be constructed even from training sets as small as 24 samples. Informed sample-selection techniques based on prediction error and model variance clearly outperform uninformed methods, such as sequential or random selection.@en

Virtually all dynamic system control methods benefit from the availability of an accurate mathematical model of the system. This includes also methods like reinforcement learning, which can be vastly sped up and made safer by using a dynamic system model. However, obtaining a sufficient amount of informative data for constructing dynamic models can be difficult. Consequently, standard data-driven model learning techniques using small data sets that do not cover all important properties of the system yield models that are partly incorrect, for instance, in terms of their steady-state characteristics or local behavior. However, often some knowledge about the desired physical properties of the model is available. Recently, several symbolic regression approaches making use of such knowledge to compensate for data insufficiency were proposed. Therefore, this knowledge should be incorporated into the model learning process to compensate for data insufficiency. In this paper, we consider a multi-objective symbolic regression method that optimizes models with respect to their training error and the measure of how well they comply with the desired physical properties. We propose an extension to the existing algorithm that helps generate a diverse set of high-quality models. Further, we propose a method for selecting a single final model out of the pool of candidate output models. We experimentally demonstrate the approach on three real systems: the TurtleBot 2 mobile robot, the Parrot Bebop 2 drone and the magnetic manipulation system. The results show that the proposed model-learning algorithm yields accurate models that are physically justified. The improvement in terms of the model's compliance with prior knowledge over the models obtained when no prior knowledge was involved in the learning process is of several orders of magnitude.@en

Visual navigation is essential for many applications in robotics, from manipulation, through mobile robotics to automated driving. Deep reinforcement learning (DRL) provides an elegant map-free approach integrating image processing, localization, and planning in one module, which can be trained and therefore optimized for a given environment. However, to date, DRL-based visual navigation was validated exclusively in simulation, where the simulator provides information that is not available in the real world, e.g., the robot's position or segmentation masks. This precludes the use of the learned policy on a real robot. Therefore, we present a novel approach that enables a direct deployment of the trained policy on real robots. We have designed a new powerful simulator capable of domain randomization. To facilitate the training, we propose visual auxiliary tasks and a tailored reward scheme. The policy is fine-tuned on images collected from real-world environments. We have evaluated the method on a mobile robot in a real office environment. The training took approximately 30 hours on a single GPU. In 30 navigation experiments, the robot reached a 0.3-meter neighbourhood of the goal in more than 86.7% of cases. This result makes the proposed method directly applicable to tasks like mobile manipulation.@en

Autonomous mobile robots are becoming increasingly important in many industrial and domestic environments. Dealing with unforeseen situations is a difficult problem that must be tackled to achieve long-term robot autonomy. In vision-based localization and navigation methods, one of the major issues is the scene dynamics. The autonomous operation of the robot may become unreliable if the changes occurring in dynamic environments are not detected and managed. Moving chairs, opening and closing doors or windows, replacing objects and other changes make many conventional methods fail. To deal with these challenges, we present a novel method for change detection based on weighted local visual features. The core idea of the algorithm is to distinguish the valuable information in stable regions of the scene from the potentially misleading information in the regions that are changing. We evaluate the change detection algorithm in a visual localization framework based on feature matching by performing a series of long-term localization experiments in various real-world environments. The results show that the change detection method yields an improvement in the localization accuracy, compared to the baseline method without change detection. In addition, an experimental evaluation on a public long-term localization data set with more than 10000 images reveals that the proposed method outperforms two alternative localization methods on images recorded several months after the initial mapping.@en

Developing mathematical models of dynamic systems is central to many disciplines of engineering and science. Models facilitate simulations, analysis of the system's behavior, decision making and design of automatic control algorithms. Even inherently model-free control techniques such as reinforcement learning (RL) have been shown to benefit from the use of models, typically learned online. Any model construction method must address the tradeoff between the accuracy of the model and its complexity, which is difficult to strike. In this paper, we propose to employ symbolic regression (SR) to construct parsimonious process models described by analytic equations. We have equipped our method with two different state-of-the-art SR algorithms which automatically search for equations that fit the measured data: Single Node Genetic Programming (SNGP) and Multi-Gene Genetic Programming (MGGP). In addition to the standard problem formulation in the state-space domain, we show how the method can also be applied to input–output models of the NARX (nonlinear autoregressive with exogenous input) type. We present the approach on three simulated examples with up to 14-dimensional state space: an inverted pendulum, a mobile robot, and a bipedal walking robot. A comparison with deep neural networks and local linear regression shows that SR in most cases outperforms these commonly used alternative methods. We demonstrate on a real pendulum system that the analytic model found enables a RL controller to successfully perform the swing-up task, based on a model constructed from only 100 data samples.@en

In symbolic regression, the search for analytic models is typically driven purely by the prediction error observed on the training data samples. However, when the data samples do not sufficiently cover the input space, the prediction error does not provide sufficient guidance toward desired models. Standard symbolic regression techniques then yield models that are partially incorrect, for instance, in terms of their steady-state characteristics or local behavior. If these properties were considered already during the search process, more accurate and relevant models could be produced. We propose a multi-objective symbolic regression approach that is driven by both the training data and the prior knowledge of the properties the desired model should manifest. The properties given in the form of formal constraints are internally represented by a set of discrete data samples on which candidate models are exactly checked. The proposed approach was experimentally evaluated on three test problems with results clearly demonstrating its capability to evolve realistic models that fit the training data well while complying with the prior knowledge of the desired model characteristics at the same time. It outperforms standard symbolic regression by several orders of magnitude in terms of the mean squared deviation from a reference model.@en

Autonomous mobile robots are becoming increasingly important in many industrial and domestic environments. Dealing with unforeseen situations is a difficult problem that must be tackled in order to move closer to the ultimate goal of life-long autonomy. In computer vision-based methods employed on mobile robots, such as localization or navigation, one of the major issues is the dynamics of the scenes. The autonomous operation of the robot may become unreliable if the changes that are common in dynamic environments are not detected and managed. Moving chairs, opening and closing doors or windows, replacing objects on the desks and other changes make many conventional methods fail. To deal with that, we present a novel method for change detection based on the similarity of local visual features. The core idea of the algorithm is to distinguish important stable regions of the scene from the regions that are changing. To evaluate the change detection algorithm, we have designed a simple visual localization framework based on feature matching and we have performed a series of real-world localization experiments. The results have shown that the change detection method substantially improves the accuracy of the robot localization, compared to using the baseline localization method without change detection.@en

Relying on static representations of the environment limits the use of mapping methods in most real-world tasks. Real-world environments are dynamic and undergo changes that need to be handled through map adaptation. In this work, an object-based pose graph is proposed to solve the problem of mapping in indoor dynamic environments with mobile robots. In contrast to state-of-The art methods where binary classifications between movable and static objects are used, we propose a new method to capture the probability of different objects over time. Object probability represents how likely it is to find a specific object in its previous location and it gives a quantification of how movable specific objects are. In addition, grouping object probabilities according to object class allows us to evaluate the movability of different object classes. We validate our object-based pose graph in real-world dynamic environments. Results in mapping and map adaptation with a real robot show efficient map maintenance through several mapping sessions and results in object classification according to movability show an improvement compared to binary classification. @en

Deep reinforcement learning (RL) has been successfully applied to a variety of game-like environments. However, the application of deep RL to visual navigation with realistic environments is a challenging task. We propose a novel learning architecture capable of navigating an agent, e.g. a mobile robot, to a target given by an image. To achieve this, we have extended the batched A2C algorithm with auxiliary tasks designed to improve visual navigation performance. We propose three additional auxiliary tasks: predicting the segmentation of the observation image and of the target image and predicting the depth-map. These tasks enable the use of supervised learning to pre-train a major part of the network and to reduce the number of training steps substantially. The training performance has been further improved by increasing the environment complexity gradually over time. An efficient neural network structure is proposed, which is capable of learning for multiple targets in multiple environments. Our method navigates in continuous state spaces and on the AI2-THOR environment simulator surpasses the performance of state-of-the-art goal-oriented visual navigation methods from the literature.@en

Reinforcement learning (RL) is a suitable approach for controlling systems with unknown or time-varying dynamics. RL in principle does not require a model of the system, but before it learns an acceptable policy, it needs many unsuccessful trials, which real robots usually cannot withstand. It is well known that RL can be sped up and made safer by using models learned online. In this paper, we propose to use symbolic regression to construct compact, parsimonious models described by analytic equations, which are suitable for realtime robot control. Single node genetic programming (SNGP) is employed as a tool to automatically search for equations fitting the available data. We demonstrate the approach on two benchmark examples: a simulated mobile robot and the pendulum swing-up problem; the latter both in simulations and real-time experiments. The results show that through this approach we can find accurate models even for small batches of training data. Based on the symbolic model found, RL can control the system well@en

The ability to search for objects is a precondition for various robotic tasks. In this paper, we address the problem of finding objects in partially known indoor environments. Using the knowledge of the floor plan and the mapped objects, we consider object-object and object-room co-occurrences as prior information for identifying promising locations where an unmapped object can be present. We propose an efficient search strategy that determines the best pose of the robot based on the analysis of the candidate locations. We optimize the probability of finding the target object and the distance travelled through a cost function.To evaluate our method, several experiments in simulated and real-world environments were performed. The results show that the robot successfully finds the target object in the environment while covering only a small portion of the search space. The real-world experiments with the TurtleBot 2 mobile robot validate the proposed approach and demonstrate that the method performs well also in real environments.@en

The ability to search for objects is a precondition for various robotic tasks. In this paper, we address the problem of finding objects in partially known indoor environments. Using the knowledge of the floor plan and the mapped objects, we consider object-object and object-room co-occurrences as prior information for identifying promising locations where an unmapped object can be present. We propose an efficient search strategy that determines the best pose of the robot based on the analysis of the candidate locations. We optimize the probability of finding the target object and the distance travelled through a cost function.To evaluate our method, several experiments in simulated and real-world environments were performed. The results show that the robot successfully finds the target object in the environment while covering only a small portion of the search space. The real-world experiments with the TurtleBot 2 mobile robot validate the proposed approach and demonstrate that the method performs well also in real environments.@en

Virtually all robot control methods benefit from the availability of an accurate mathematical model of the robot. However, obtaining a sufficient amount of informative data for constructing dynamic models can be difficult, especially when the models are to be learned during robot deployment. Under such circumstances, standard data-driven model learning techniques often yield models that do not comply with the physics of the robot. We extend a symbolic regression algorithm based on Single Node Genetic Programming by including the prior model information into the model construction process. In this way, symbolic regression automatically builds models that compensate for theoretical or empirical model deficiencies. We experimentally demonstrate the approach on two real-world systems: the TurtleBot 2 mobile robot and the Parrot Bebop 2 drone. The results show that the proposed model-learning algorithm produces realistic models that fit well the training data even when using small training sets. Passing the prior model information to the algorithm significantly improves the model accuracy while speeding up the search.@en

Virtually all robot control methods benefit from the availability of an accurate mathematical model of the robot. However, obtaining a sufficient amount of informative data for constructing dynamic models can be difficult, especially when the models are to be learned during robot deployment. Under such circumstances, standard data-driven model learning techniques often yield models that do not comply with the physics of the robot. We extend a symbolic regression algorithm based on Single Node Genetic Programming by including the prior model information into the model construction process. In this way, symbolic regression automatically builds models that compensate for theoretical or empirical model deficiencies. We experimentally demonstrate the approach on two real-world systems: the TurtleBot 2 mobile robot and the Parrot Bebop 2 drone. The results show that the proposed model-learning algorithm produces realistic models that fit well the training data even when using small training sets. Passing the prior model information to the algorithm significantly improves the model accuracy while speeding up the search.@en

It is well known that reinforcement learning (RL) can benefit from the use of a dynamic prediction model which is learned on data samples collected online from the process to be controlled. Most RL algorithms are formulated in the state-space domain and use state-space models. However, learning state-space models is difficult, mainly because in the vast majority of problems the full state cannot be measured on the system or reconstructed from the measurements. To circumvent this limitation, we propose to use input-output models of the NARX (nonlinear autoregressive with exogenous input) type. Symbolic regression is employed to construct parsimonious models and the corresponding value functions. Thanks to this approach, we can learn accurate models and compute optimal policies even from small amounts of training data. We demonstrate the approach on two simulated examples, a hopping robot and a 1-DOF robot arm, and on a real inverted pendulum system. Results show that our proposed method can reliably determine a good control policy based on a symbolic input-output process model and value function.@en