Ruckig calculates a time-optimal trajectory given a target waypoint with position, velocity, and acceleration starting from any initial state limited by velocity, acceleration, and jerk constraints. Ruckig is a more powerful and open-source alternative to the Reflexxes Type IV library. In fact, Ruckig is the first Type V trajectory generator and even supports directional velocity and acceleration limits, while also being faster on top. For robotics and machining applications, Ruckig allows both instant reactions to unforeseen events as well as simple offline trajectory planning.

More information can be found in the corresponding paper Jerk-limited Real-time Trajectory Generation with Arbitrary Target States, accepted for the Robotics: Science and Systems (RSS), 2021 conference.

Installation

Ruckig has no dependencies (except for testing). To build Ruckig using CMake, just run

mkdir -p build
cd build
cmake -DCMAKE_BUILD_TYPE=Release ..
make

To install Ruckig in a system-wide directory, use (sudo) make install. An example of using Ruckig in your CMake project is given by examples/CMakeLists.txt. However, you can also include Ruckig as a directory within your project and call add_subdirectory(ruckig) in your parent CMakeLists.txt.

Ruckig is also available as a Python module, in particular for development or debugging purposes. It can be installed from PyPI via

pip install ruckig

When using CMake, the Python module can be built using the BUILD_PYTHON_MODULE flag. If you're only interested in the Python module (and not in the C++ library), you can build and install Ruckig via pip install ..

Tutorial

Furthermore, a tutorial will explain the basics to include online generated trajectories within your application. A working example can be found in the examples directory. A time-optimal trajectory for a single degree of freedom is shown in the figure below.

Waypoint-based Trajectory Generation

Ruckig provides three main interface classes: the Ruckig, the InputParameter, and the OutputParameter class.

First, you'll need to create a Ruckig instance with the number of DoFs as a template parameter, and the control cycle (e.g. in seconds) in the constructor.

Ruckig<6> ruckig {0.001}; // Number DoFs; control cycle in [s]

The input type has 3 blocks of data: the current state, the target state and the corresponding kinematic limits.

InputParameter<6> input; // Number DoFs
input.current_position = {0.2, ...};
input.current_velocity = {0.1, ...};
input.current_acceleration = {0.1, ...};
input.target_position = {0.5, ...};
input.target_velocity = {-0.1, ...};
input.target_acceleration = {0.2, ...};
input.max_velocity = {0.4, ...};
input.max_acceleration = {1.0, ...};
input.max_jerk = {4.0, ...};

OutputParameter<6> output; // Number DoFs

Given all input and output resources, we can iterate over the trajectory at each discrete time step. For most applications, this loop must run within a real-time thread and controls the actual hardware.

while (ruckig.update(input, output) == Result::Working) {
  // Make use of the new state here!

  input.current_position = output.new_position;
  input.current_velocity = output.new_velocity;
  input.current_acceleration = output.new_acceleration;
}

During your update step, you'll need to copy the new kinematic state into the current state. If the current state is not the expected, pre-calculated trajectory, ruckig will calculate a new trajectory with the novel input. When the trajectory has reached the target state, the update function will return Result::Finished.

Input Parameter

To go into more detail, the InputParameter type has following members:

using Vector = std::array<double, DOFs>;

Vector current_position;
Vector current_velocity; // Initialized to zero
Vector current_acceleration; // Initialized to zero

Vector target_position;
Vector target_velocity; // Initialized to zero
Vector target_acceleration; // Initialized to zero

Vector max_velocity;
Vector max_acceleration;
Vector max_jerk;

std::optional<Vector> min_velocity; // If not given, the negative maximum velocity will be used.
std::optional<Vector> min_acceleration; // If not given, the negative maximum acceleration will be used.

std::array<bool, DOFs> enabled; // Initialized to true
std::optional<double> minimum_duration;

Interface interface; // The default position interface controls the full kinematic state.
Synchronization synchronization; // Synchronization behavior of multiple DoFs
DurationDiscretization duration_discretization; // Whether the duration should be a discrete multiple of the control cycle (off by default)

Members are implemented using the C++ standard array and optional type. Note that there are range constraints due to numerical reasons, see below for more details. To check the input before a calculation step, the ruckig.validate_input(input) method returns false if an input is not valid. Of course, the target state needs to be within the given kinematic limits. Additionally, the target acceleration needs to fulfil

Abs(target_acceleration) <= Sqrt(2 * max_jerk * (max_velocity - Abs(target_velocity)))

If a DoF is not enabled, it will be ignored in the calculation. A minimum duration can be optionally given. Furthermore, the minimum velocity and acceleration can be specified. If it is not given, the negative maximum velocity or acceleration will be used (similar to the jerk limit). For example, this might be useful in human robot collaboration settings with a different velocity limit towards a human. Or, the dynamic limits at a given configuration of the robot can be approximated much better with different acceleration limits.

Furthermore, there are some options for advanced functionality, e.g.:

• for different synchronization behavior (i.a. phase or time synchonization).
• for the control interface (position or velocity control).
• for discrete trajectory durations.

We refer to the API documentation of the enumerations within the ruckig namespace for all available options.

Result Type

The update function of the Ruckig class returns a Result type that indicates the current state of the algorithm. Currently, this can either be working, finished if the trajectory has finished, or an error type if something went wrong during calculation. The result type can be compared as a standard integer.

Output Parameter

The output class gives the new kinematic state of the trajectory.

Vector new_position;
Vector new_velocity;
Vector new_acceleration;

bool new_calculation; // Whether a new calculation was performed in the last cycle
double calculation_duration; // Duration of the calculation in the last cycle [µs]

Trajectory trajectory; // The current trajectory
double time; // The current, auto-incremented time. Reset to 0 at a new calculation.

Moreover, the trajectory class has a range of useful parameters and methods.

double duration; // Duration of the trajectory
std::array<double, DOFs> independent_min_durations; // Time-optimal profile for each independent DoF

<...> at_time(double time); // Get the kinematic state of the trajectory at a given time
<...> get_position_extrema(); // Returns information about the position extrema and their times

Again, we refer to the API documentation for the exact signatures.

Tests and Numerical Stability

The current test suite validates over 5.000.000.000 random trajectories. The numerical exactness is tested for the final position and final velocity to be within 1e-8, for the velocity, acceleration and jerk limit to be within 1e-12, and for the final acceleration as well to be within a numerical error of 1e-12. The maximal supported trajectory duration is 7e3, which sounds short but should suffice for most applications seeking for time-optimality. Note that Ruckig will also output values outside of this range, there is however no guarantee for correctness.

Benchmark

We find that Ruckig is around twice as fast as Reflexxes Type IV and well-suited for control cycles as low as half a millisecond.

Development

Ruckig is written in C++17. It is continuously tested on ubuntu-latest, macos-latest, and windows-latest against following versions

• Doctest v2.4 (only for testing)
• Pybind11 v2.6 (only for python wrapper)

Citation

@article{berscheid2021jerk,
  title={Jerk-limited Real-time Trajectory Generation with Arbitrary Target States},
  author={Berscheid, Lars and Kr{\"o}ger, Torsten},
  journal={Robotics: Science and Systems XVII (accepted)},
  year={2021}
}