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ar_basalt/thirdparty/basalt-headers/include/basalt/spline/so3_spline.h
Ivan 6dc0eb0fcf v01
2022-04-05 11:42:28 +03:00

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/**
BSD 3-Clause License
This file is part of the Basalt project.
https://gitlab.com/VladyslavUsenko/basalt-headers.git
Copyright (c) 2019, Vladyslav Usenko and Nikolaus Demmel.
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
* Redistributions of source code must retain the above copyright notice, this
list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above copyright notice,
this list of conditions and the following disclaimer in the documentation
and/or other materials provided with the distribution.
* Neither the name of the copyright holder nor the names of its
contributors may be used to endorse or promote products derived from
this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@file
@brief Uniform cumulative B-spline for SO(3)
*/
#pragma once
#include <basalt/spline/spline_common.h>
#include <basalt/utils/assert.h>
#include <basalt/utils/sophus_utils.hpp>
#include <Eigen/Dense>
#include <sophus/so3.hpp>
#include <array>
namespace basalt {
/// @brief Uniform cummulative B-spline for SO(3) of order N
///
/// For example, in the particular case scalar values and order N=5, for a time
/// \f$t \in [t_i, t_{i+1})\f$ the value of \f$p(t)\f$ depends only on 5 control
/// points at \f$[t_i, t_{i+1}, t_{i+2}, t_{i+3}, t_{i+4}]\f$. To
/// simplify calculations we transform time to uniform representation \f$s(t) =
/// (t - t_0)/\Delta t \f$, such that control points transform into \f$ s_i \in
/// [0,..,N] \f$. We define function \f$ u(t) = s(t)-s_i \f$ to be a time since
/// the start of the segment. Following the cummulative matrix representation of
/// De Boor - Cox formula, the value of the function can be evaluated as
/// follows: \f{align}{
/// R(u(t)) &= R_i
/// \prod_{j=1}^{4}{\exp(k_{j}\log{(R_{i+j-1}^{-1}R_{i+j})})},
/// \\ \begin{pmatrix} k_0 \\ k_1 \\ k_2 \\ k_3 \\ k_4 \end{pmatrix}^T &=
/// M_{c5} \begin{pmatrix} 1 \\ u \\ u^2 \\ u^3 \\ u^4
/// \end{pmatrix},
/// \f}
/// where \f$ R_{i} \in SO(3) \f$ are knots and \f$ M_{c5} \f$ is a cummulative
/// blending matrix computed using \ref computeBlendingMatrix \f{align}{
/// M_{c5} = \frac{1}{4!}
/// \begin{pmatrix} 24 & 0 & 0 & 0 & 0 \\ 23 & 4 & -6 & 4 & -1 \\ 12 & 16 & 0
/// & -8 & 3 \\ 1 & 4 & 6 & 4 & -3 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}.
/// \f}
///
/// See [[arXiv:1911.08860]](https://arxiv.org/abs/1911.08860) for more details.
template <int _N, typename _Scalar = double>
class So3Spline {
public:
static constexpr int N = _N; ///< Order of the spline.
static constexpr int DEG = _N - 1; ///< Degree of the spline.
static constexpr _Scalar NS_TO_S = 1e-9; ///< Nanosecond to second conversion
static constexpr _Scalar S_TO_NS = 1e9; ///< Second to nanosecond conversion
using MatN = Eigen::Matrix<_Scalar, _N, _N>;
using VecN = Eigen::Matrix<_Scalar, _N, 1>;
using Vec3 = Eigen::Matrix<_Scalar, 3, 1>;
using Mat3 = Eigen::Matrix<_Scalar, 3, 3>;
using SO3 = Sophus::SO3<_Scalar>;
/// @brief Struct to store the Jacobian of the spline
///
/// Since B-spline of order N has local support (only N knots infuence the
/// value) the Jacobian is zero for all knots except maximum N for value and
/// all derivatives.
struct JacobianStruct {
size_t start_idx;
std::array<Mat3, _N> d_val_d_knot;
};
/// @brief Constructor with knot interval and start time
///
/// @param[in] time_interval_ns knot time interval in nanoseconds
/// @param[in] start_time_ns start time of the spline in nanoseconds
So3Spline(int64_t time_interval_ns, int64_t start_time_ns = 0)
: dt_ns_(time_interval_ns), start_t_ns_(start_time_ns) {
pow_inv_dt_[0] = 1.0;
pow_inv_dt_[1] = S_TO_NS / dt_ns_;
pow_inv_dt_[2] = pow_inv_dt_[1] * pow_inv_dt_[1];
pow_inv_dt_[3] = pow_inv_dt_[2] * pow_inv_dt_[1];
}
/// @brief Maximum time represented by spline
///
/// @return maximum time represented by spline in nanoseconds
int64_t maxTimeNs() const {
return start_t_ns_ + (knots_.size() - N + 1) * dt_ns_ - 1;
}
/// @brief Minimum time represented by spline
///
/// @return minimum time represented by spline in nanoseconds
int64_t minTimeNs() const { return start_t_ns_; }
/// @brief Gererate random trajectory
///
/// @param[in] n number of knots to generate
/// @param[in] static_init if true the first N knots will be the same
/// resulting in static initial condition
void genRandomTrajectory(int n, bool static_init = false) {
if (static_init) {
Vec3 rnd = Vec3::Random() * M_PI;
for (int i = 0; i < N; i++) {
knots_.push_back(SO3::exp(rnd));
}
for (int i = 0; i < n - N; i++) {
knots_.push_back(knots_.back() * SO3::exp(Vec3::Random() * M_PI / 2));
}
} else {
knots_.push_back(SO3::exp(Vec3::Random() * M_PI));
for (int i = 1; i < n; i++) {
knots_.push_back(knots_.back() * SO3::exp(Vec3::Random() * M_PI / 2));
}
}
}
/// @brief Set start time for spline
///
/// @param[in] start_time_ns start time of the spline in nanoseconds
inline void setStartTimeNs(int64_t s) { start_t_ns_ = s; }
/// @brief Add knot to the end of the spline
///
/// @param[in] knot knot to add
inline void knotsPushBack(const SO3& knot) { knots_.push_back(knot); }
/// @brief Remove knot from the back of the spline
inline void knotsPopBack() { knots_.pop_back(); }
/// @brief Return the first knot of the spline
///
/// @return first knot of the spline
inline const SO3& knotsFront() const { return knots_.front(); }
/// @brief Remove first knot of the spline and increase the start time
inline void knotsPopFront() {
start_t_ns_ += dt_ns_;
knots_.pop_front();
}
/// @brief Resize containter with knots
///
/// @param[in] n number of knots
inline void resize(size_t n) { knots_.resize(n); }
/// @brief Return reference to the knot with index i
///
/// @param i index of the knot
/// @return reference to the knot
inline SO3& getKnot(int i) { return knots_[i]; }
/// @brief Return const reference to the knot with index i
///
/// @param i index of the knot
/// @return const reference to the knot
inline const SO3& getKnot(int i) const { return knots_[i]; }
/// @brief Return const reference to deque with knots
///
/// @return const reference to deque with knots
const Eigen::aligned_deque<SO3>& getKnots() const { return knots_; }
/// @brief Return time interval in nanoseconds
///
/// @return time interval in nanoseconds
int64_t getTimeIntervalNs() const { return dt_ns_; }
/// @brief Evaluate SO(3) B-spline
///
/// @param[in] time_ns time for evaluating the value of the spline in
/// nanoseconds
/// @param[out] J if not nullptr, return the Jacobian of the value with
/// respect to knots
/// @return SO(3) value of the spline
SO3 evaluate(int64_t time_ns, JacobianStruct* J = nullptr) const {
int64_t st_ns = (time_ns - start_t_ns_);
BASALT_ASSERT_STREAM(st_ns >= 0, "st_ns " << st_ns << " time_ns " << time_ns
<< " start_t_ns " << start_t_ns_);
int64_t s = st_ns / dt_ns_;
double u = double(st_ns % dt_ns_) / double(dt_ns_);
BASALT_ASSERT_STREAM(s >= 0, "s " << s);
BASALT_ASSERT_STREAM(
size_t(s + N) <= knots_.size(),
"s " << s << " N " << N << " knots.size() " << knots_.size());
VecN p;
baseCoeffsWithTime<0>(p, u);
VecN coeff = BLENDING_MATRIX * p;
SO3 res = knots_[s];
Mat3 J_helper;
if (J) {
J->start_idx = s;
J_helper.setIdentity();
}
for (int i = 0; i < DEG; i++) {
const SO3& p0 = knots_[s + i];
const SO3& p1 = knots_[s + i + 1];
SO3 r01 = p0.inverse() * p1;
Vec3 delta = r01.log();
Vec3 kdelta = delta * coeff[i + 1];
if (J) {
Mat3 Jl_inv_delta;
Mat3 Jl_k_delta;
Sophus::leftJacobianInvSO3(delta, Jl_inv_delta);
Sophus::leftJacobianSO3(kdelta, Jl_k_delta);
J->d_val_d_knot[i] = J_helper;
J_helper = coeff[i + 1] * res.matrix() * Jl_k_delta * Jl_inv_delta *
p0.inverse().matrix();
J->d_val_d_knot[i] -= J_helper;
}
res *= SO3::exp(kdelta);
}
if (J) {
J->d_val_d_knot[DEG] = J_helper;
}
return res;
}
/// @brief Evaluate rotational velocity (first time derivative) of SO(3)
/// B-spline in the body frame
/// First, let's note that for scalars \f$ k, \Delta k \f$ the following
/// holds: \f$ \exp((k+\Delta k)\phi) = \exp(k\phi)\exp(\Delta k\phi), \phi
/// \in \mathbb{R}^3\f$. This is due to the fact that rotations around the
/// same axis are commutative.
///
/// Let's take SO(3) B-spline with N=3 as an example. The evolution in time of
/// rotation from the body frame to the world frame is described with \f[
/// R_{wb}(t) = R(t) = R_i \exp(k_1(t) \log(R_{i}^{-1}R_{i+1})) \exp(k_2(t)
/// \log(R_{i+1}^{-1}R_{i+2})), \f] where \f$ k_1, k_2 \f$ are spline
/// coefficients (see detailed description of \ref So3Spline). Since
/// expressions under logmap do not depend on time we can rename them to
/// constants.
/// \f[ R(t) = R_i \exp(k_1(t) ~ d_1) \exp(k_2(t) ~ d_2). \f]
///
/// With linear approximation of the spline coefficient evolution over time
/// \f$ k_1(t_0 + \Delta t) = k_1(t_0) + k_1'(t_0)\Delta t \f$ we can write
/// \f{align}
/// R(t_0 + \Delta t) &= R_i \exp( (k_1(t_0) + k_1'(t_0) \Delta t) ~ d_1)
/// \exp((k_2(t_0) + k_2'(t_0) \Delta t) ~ d_2)
/// \\ &= R_i \exp(k_1(t_0) ~ d_1) \exp(k_1'(t_0)~ d_1 \Delta t )
/// \exp(k_2(t_0) ~ d_2) \exp(k_2'(t_0) ~ d_2 \Delta t )
/// \\ &= R_i \exp(k_1(t_0) ~ d_1)
/// \exp(k_2(t_0) ~ d_2) \exp(R_{a}^T k_1'(t_0)~ d_1 \Delta t )
/// \exp(k_2'(t_0) ~ d_2 \Delta t )
/// \\ &= R_i \exp(k_1(t_0) ~ d_1)
/// \exp(k_2(t_0) ~ d_2) \exp((R_{a}^T k_1'(t_0)~ d_1 +
/// k_2'(t_0) ~ d_2) \Delta t )
/// \\ &= R(t_0) \exp((R_{a}^T k_1'(t_0)~ d_1 +
/// k_2'(t_0) ~ d_2) \Delta t )
/// \\ &= R(t_0) \exp( \omega \Delta t ),
/// \f} where \f$ \Delta t \f$ is small, \f$ R_{a} \in SO(3) = \exp(k_2(t_0) ~
/// d_2) \f$ and \f$ \omega \f$ is the rotational velocity in the body frame.
/// More explicitly we have the formula for rotational velocity in the body
/// frame \f[ \omega = R_{a}^T k_1'(t_0)~ d_1 + k_2'(t_0) ~ d_2. \f]
/// Derivatives of spline coefficients can be computed with \ref
/// baseCoeffsWithTime similar to \ref RdSpline (detailed description). With
/// the recursive formula computations generalize to different orders of
/// spline N.
///
/// @param[in] time_ns time for evaluating velocity of the spline in
/// nanoseconds
/// @return rotational velocity (3x1 vector)
Vec3 velocityBody(int64_t time_ns) const {
int64_t st_ns = (time_ns - start_t_ns_);
BASALT_ASSERT_STREAM(st_ns >= 0, "st_ns " << st_ns << " time_ns " << time_ns
<< " start_t_ns " << start_t_ns_);
int64_t s = st_ns / dt_ns_;
double u = double(st_ns % dt_ns_) / double(dt_ns_);
BASALT_ASSERT_STREAM(s >= 0, "s " << s);
BASALT_ASSERT_STREAM(
size_t(s + N) <= knots_.size(),
"s " << s << " N " << N << " knots.size() " << knots_.size());
VecN p;
baseCoeffsWithTime<0>(p, u);
VecN coeff = BLENDING_MATRIX * p;
baseCoeffsWithTime<1>(p, u);
VecN dcoeff = pow_inv_dt_[1] * BLENDING_MATRIX * p;
Vec3 rot_vel;
rot_vel.setZero();
for (int i = 0; i < DEG; i++) {
const SO3& p0 = knots_[s + i];
const SO3& p1 = knots_[s + i + 1];
SO3 r01 = p0.inverse() * p1;
Vec3 delta = r01.log();
rot_vel = SO3::exp(-delta * coeff[i + 1]) * rot_vel;
rot_vel += delta * dcoeff[i + 1];
}
return rot_vel;
}
/// @brief Evaluate rotational velocity (first time derivative) of SO(3)
/// B-spline in the body frame
///
/// @param[in] time_ns time for evaluating velocity of the spline in
/// nanoseconds
/// @param[out] J if not nullptr, return the Jacobian of the rotational
/// velocity in body frame with respect to knots
/// @return rotational velocity (3x1 vector)
Vec3 velocityBody(int64_t time_ns, JacobianStruct* J) const {
int64_t st_ns = (time_ns - start_t_ns_);
BASALT_ASSERT_STREAM(st_ns >= 0, "st_ns " << st_ns << " time_ns " << time_ns
<< " start_t_ns " << start_t_ns_);
int64_t s = st_ns / dt_ns_;
double u = double(st_ns % dt_ns_) / double(dt_ns_);
BASALT_ASSERT_STREAM(s >= 0, "s " << s);
BASALT_ASSERT_STREAM(
size_t(s + N) <= knots_.size(),
"s " << s << " N " << N << " knots.size() " << knots_.size());
VecN p;
baseCoeffsWithTime<0>(p, u);
VecN coeff = BLENDING_MATRIX * p;
baseCoeffsWithTime<1>(p, u);
VecN dcoeff = pow_inv_dt_[1] * BLENDING_MATRIX * p;
Vec3 delta_vec[DEG];
Mat3 R_tmp[DEG];
SO3 accum;
SO3 exp_k_delta[DEG];
Mat3 Jr_delta_inv[DEG];
Mat3 Jr_kdelta[DEG];
for (int i = DEG - 1; i >= 0; i--) {
const SO3& p0 = knots_[s + i];
const SO3& p1 = knots_[s + i + 1];
SO3 r01 = p0.inverse() * p1;
delta_vec[i] = r01.log();
Sophus::rightJacobianInvSO3(delta_vec[i], Jr_delta_inv[i]);
Jr_delta_inv[i] *= p1.inverse().matrix();
Vec3 k_delta = coeff[i + 1] * delta_vec[i];
Sophus::rightJacobianSO3(-k_delta, Jr_kdelta[i]);
R_tmp[i] = accum.matrix();
exp_k_delta[i] = Sophus::SO3d::exp(-k_delta);
accum *= exp_k_delta[i];
}
Mat3 d_vel_d_delta[DEG];
d_vel_d_delta[0] = dcoeff[1] * R_tmp[0] * Jr_delta_inv[0];
Vec3 rot_vel = delta_vec[0] * dcoeff[1];
for (int i = 1; i < DEG; i++) {
d_vel_d_delta[i] =
R_tmp[i - 1] * SO3::hat(rot_vel) * Jr_kdelta[i] * coeff[i + 1] +
R_tmp[i] * dcoeff[i + 1];
d_vel_d_delta[i] *= Jr_delta_inv[i];
rot_vel = exp_k_delta[i] * rot_vel + delta_vec[i] * dcoeff[i + 1];
}
if (J) {
J->start_idx = s;
for (int i = 0; i < N; i++) {
J->d_val_d_knot[i].setZero();
}
for (int i = 0; i < DEG; i++) {
J->d_val_d_knot[i] -= d_vel_d_delta[i];
J->d_val_d_knot[i + 1] += d_vel_d_delta[i];
}
}
return rot_vel;
}
/// @brief Evaluate rotational acceleration (second time derivative) of SO(3)
/// B-spline in the body frame
///
/// @param[in] time_ns time for evaluating acceleration of the spline in
/// nanoseconds
/// @param[out] vel_body if not nullptr, return the rotational velocity in the
/// body frame (3x1 vector) (side computation)
/// @return rotational acceleration (3x1 vector)
Vec3 accelerationBody(int64_t time_ns, Vec3* vel_body = nullptr) const {
int64_t st_ns = (time_ns - start_t_ns_);
BASALT_ASSERT_STREAM(st_ns >= 0, "st_ns " << st_ns << " time_ns " << time_ns
<< " start_t_ns " << start_t_ns_);
int64_t s = st_ns / dt_ns_;
double u = double(st_ns % dt_ns_) / double(dt_ns_);
BASALT_ASSERT_STREAM(s >= 0, "s " << s);
BASALT_ASSERT_STREAM(
size_t(s + N) <= knots_.size(),
"s " << s << " N " << N << " knots.size() " << knots_.size());
VecN p;
baseCoeffsWithTime<0>(p, u);
VecN coeff = BLENDING_MATRIX * p;
baseCoeffsWithTime<1>(p, u);
VecN dcoeff = pow_inv_dt_[1] * BLENDING_MATRIX * p;
baseCoeffsWithTime<2>(p, u);
VecN ddcoeff = pow_inv_dt_[2] * BLENDING_MATRIX * p;
SO3 r_accum;
Vec3 rot_vel;
rot_vel.setZero();
Vec3 rot_accel;
rot_accel.setZero();
for (int i = 0; i < DEG; i++) {
const SO3& p0 = knots_[s + i];
const SO3& p1 = knots_[s + i + 1];
SO3 r01 = p0.inverse() * p1;
Vec3 delta = r01.log();
SO3 rot = SO3::exp(-delta * coeff[i + 1]);
rot_vel = rot * rot_vel;
Vec3 vel_current = dcoeff[i + 1] * delta;
rot_vel += vel_current;
rot_accel = rot * rot_accel;
rot_accel += ddcoeff[i + 1] * delta + rot_vel.cross(vel_current);
}
if (vel_body) {
*vel_body = rot_vel;
}
return rot_accel;
}
/// @brief Evaluate rotational acceleration (second time derivative) of SO(3)
/// B-spline in the body frame
///
/// @param[in] time_ns time for evaluating acceleration of the spline in
/// nanoseconds
/// @param[out] J_accel if not nullptr, return the Jacobian of the rotational
/// acceleration in body frame with respect to knots
/// @param[out] vel_body if not nullptr, return the rotational velocity in the
/// body frame (3x1 vector) (side computation)
/// @param[out] J_vel if not nullptr, return the Jacobian of the rotational
/// velocity in the body frame (side computation)
/// @return rotational acceleration (3x1 vector)
Vec3 accelerationBody(int64_t time_ns, JacobianStruct* J_accel,
Vec3* vel_body = nullptr,
JacobianStruct* J_vel = nullptr) const {
BASALT_ASSERT(J_accel);
int64_t st_ns = (time_ns - start_t_ns_);
BASALT_ASSERT_STREAM(st_ns >= 0, "st_ns " << st_ns << " time_ns " << time_ns
<< " start_t_ns " << start_t_ns_);
int64_t s = st_ns / dt_ns_;
double u = double(st_ns % dt_ns_) / double(dt_ns_);
BASALT_ASSERT_STREAM(s >= 0, "s " << s);
BASALT_ASSERT_STREAM(
size_t(s + N) <= knots_.size(),
"s " << s << " N " << N << " knots.size() " << knots_.size());
VecN p;
baseCoeffsWithTime<0>(p, u);
VecN coeff = BLENDING_MATRIX * p;
baseCoeffsWithTime<1>(p, u);
VecN dcoeff = pow_inv_dt_[1] * BLENDING_MATRIX * p;
baseCoeffsWithTime<2>(p, u);
VecN ddcoeff = pow_inv_dt_[2] * BLENDING_MATRIX * p;
Vec3 delta_vec[DEG];
Mat3 exp_k_delta[DEG];
Mat3 Jr_delta_inv[DEG];
Mat3 Jr_kdelta[DEG];
Vec3 rot_vel;
rot_vel.setZero();
Vec3 rot_accel;
rot_accel.setZero();
Vec3 rot_vel_arr[DEG];
Vec3 rot_accel_arr[DEG];
for (int i = 0; i < DEG; i++) {
const SO3& p0 = knots_[s + i];
const SO3& p1 = knots_[s + i + 1];
SO3 r01 = p0.inverse() * p1;
delta_vec[i] = r01.log();
Sophus::rightJacobianInvSO3(delta_vec[i], Jr_delta_inv[i]);
Jr_delta_inv[i] *= p1.inverse().matrix();
Vec3 k_delta = coeff[i + 1] * delta_vec[i];
Sophus::rightJacobianSO3(-k_delta, Jr_kdelta[i]);
exp_k_delta[i] = Sophus::SO3d::exp(-k_delta).matrix();
rot_vel = exp_k_delta[i] * rot_vel;
Vec3 vel_current = dcoeff[i + 1] * delta_vec[i];
rot_vel += vel_current;
rot_accel = exp_k_delta[i] * rot_accel;
rot_accel += ddcoeff[i + 1] * delta_vec[i] + rot_vel.cross(vel_current);
rot_vel_arr[i] = rot_vel;
rot_accel_arr[i] = rot_accel;
}
Mat3 d_accel_d_delta[DEG];
Mat3 d_vel_d_delta[DEG];
d_vel_d_delta[DEG - 1] = coeff[DEG] * exp_k_delta[DEG - 1] *
SO3::hat(rot_vel_arr[DEG - 2]) *
Jr_kdelta[DEG - 1] +
Mat3::Identity() * dcoeff[DEG];
d_accel_d_delta[DEG - 1] =
coeff[DEG] * exp_k_delta[DEG - 1] * SO3::hat(rot_accel_arr[DEG - 2]) *
Jr_kdelta[DEG - 1] +
Mat3::Identity() * ddcoeff[DEG] +
dcoeff[DEG] * (SO3::hat(rot_vel_arr[DEG - 1]) -
SO3::hat(delta_vec[DEG - 1]) * d_vel_d_delta[DEG - 1]);
Mat3 pj;
pj.setIdentity();
Vec3 sj;
sj.setZero();
for (int i = DEG - 2; i >= 0; i--) {
sj += dcoeff[i + 2] * pj * delta_vec[i + 1];
pj *= exp_k_delta[i + 1];
d_vel_d_delta[i] = Mat3::Identity() * dcoeff[i + 1];
if (i >= 1) {
d_vel_d_delta[i] += coeff[i + 1] * exp_k_delta[i] *
SO3::hat(rot_vel_arr[i - 1]) * Jr_kdelta[i];
}
d_accel_d_delta[i] =
Mat3::Identity() * ddcoeff[i + 1] +
dcoeff[i + 1] * (SO3::hat(rot_vel_arr[i]) -
SO3::hat(delta_vec[i]) * d_vel_d_delta[i]);
if (i >= 1) {
d_accel_d_delta[i] += coeff[i + 1] * exp_k_delta[i] *
SO3::hat(rot_accel_arr[i - 1]) * Jr_kdelta[i];
}
d_vel_d_delta[i] = pj * d_vel_d_delta[i];
d_accel_d_delta[i] =
pj * d_accel_d_delta[i] - SO3::hat(sj) * d_vel_d_delta[i];
}
if (J_vel) {
J_vel->start_idx = s;
for (int i = 0; i < N; i++) {
J_vel->d_val_d_knot[i].setZero();
}
for (int i = 0; i < DEG; i++) {
Mat3 val = d_vel_d_delta[i] * Jr_delta_inv[i];
J_vel->d_val_d_knot[i] -= val;
J_vel->d_val_d_knot[i + 1] += val;
}
}
if (J_accel) {
J_accel->start_idx = s;
for (int i = 0; i < N; i++) {
J_accel->d_val_d_knot[i].setZero();
}
for (int i = 0; i < DEG; i++) {
Mat3 val = d_accel_d_delta[i] * Jr_delta_inv[i];
J_accel->d_val_d_knot[i] -= val;
J_accel->d_val_d_knot[i + 1] += val;
}
}
if (vel_body) {
*vel_body = rot_vel;
}
return rot_accel;
}
/// @brief Evaluate rotational jerk (third time derivative) of SO(3)
/// B-spline in the body frame
///
/// @param[in] time_ns time for evaluating jerk of the spline in
/// nanoseconds
/// @param[out] vel_body if not nullptr, return the rotational velocity in the
/// body frame (3x1 vector) (side computation)
/// @param[out] accel_body if not nullptr, return the rotational acceleration
/// in the body frame (3x1 vector) (side computation)
/// @return rotational jerk (3x1 vector)
Vec3 jerkBody(int64_t time_ns, Vec3* vel_body = nullptr,
Vec3* accel_body = nullptr) const {
int64_t st_ns = (time_ns - start_t_ns_);
BASALT_ASSERT_STREAM(st_ns >= 0, "st_ns " << st_ns << " time_ns " << time_ns
<< " start_t_ns " << start_t_ns_);
int64_t s = st_ns / dt_ns_;
double u = double(st_ns % dt_ns_) / double(dt_ns_);
BASALT_ASSERT_STREAM(s >= 0, "s " << s);
BASALT_ASSERT_STREAM(
size_t(s + N) <= knots_.size(),
"s " << s << " N " << N << " knots.size() " << knots_.size());
VecN p;
baseCoeffsWithTime<0>(p, u);
VecN coeff = BLENDING_MATRIX * p;
baseCoeffsWithTime<1>(p, u);
VecN dcoeff = pow_inv_dt_[1] * BLENDING_MATRIX * p;
baseCoeffsWithTime<2>(p, u);
VecN ddcoeff = pow_inv_dt_[2] * BLENDING_MATRIX * p;
baseCoeffsWithTime<3>(p, u);
VecN dddcoeff = pow_inv_dt_[3] * BLENDING_MATRIX * p;
Vec3 rot_vel;
rot_vel.setZero();
Vec3 rot_accel;
rot_accel.setZero();
Vec3 rot_jerk;
rot_jerk.setZero();
for (int i = 0; i < DEG; i++) {
const SO3& p0 = knots_[s + i];
const SO3& p1 = knots_[s + i + 1];
SO3 r01 = p0.inverse() * p1;
Vec3 delta = r01.log();
SO3 rot = SO3::exp(-delta * coeff[i + 1]);
rot_vel = rot * rot_vel;
Vec3 vel_current = dcoeff[i + 1] * delta;
rot_vel += vel_current;
rot_accel = rot * rot_accel;
Vec3 rot_vel_cross_vel_current = rot_vel.cross(vel_current);
rot_accel += ddcoeff[i + 1] * delta + rot_vel_cross_vel_current;
rot_jerk = rot * rot_jerk;
rot_jerk += dddcoeff[i + 1] * delta +
(ddcoeff[i + 1] * rot_vel + 2 * dcoeff[i + 1] * rot_accel -
dcoeff[i + 1] * rot_vel_cross_vel_current)
.cross(delta);
}
if (vel_body) {
*vel_body = rot_vel;
}
if (accel_body) {
*accel_body = rot_accel;
}
return rot_jerk;
}
EIGEN_MAKE_ALIGNED_OPERATOR_NEW
protected:
/// @brief Vector of derivatives of time polynomial.
///
/// Computes a derivative of \f$ \begin{bmatrix}1 & t & t^2 & \dots &
/// t^{N-1}\end{bmatrix} \f$ with repect to time. For example, the first
/// derivative would be \f$ \begin{bmatrix}0 & 1 & 2 t & \dots & (N-1)
/// t^{N-2}\end{bmatrix} \f$.
/// @param Derivative derivative to evaluate
/// @param[out] res_const vector to store the result
/// @param[in] t
template <int Derivative, class Derived>
static void baseCoeffsWithTime(const Eigen::MatrixBase<Derived>& res_const,
_Scalar t) {
EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(Derived, N);
Eigen::MatrixBase<Derived>& res =
const_cast<Eigen::MatrixBase<Derived>&>(res_const);
res.setZero();
if (Derivative < N) {
res[Derivative] = BASE_COEFFICIENTS(Derivative, Derivative);
_Scalar ti = t;
for (int j = Derivative + 1; j < N; j++) {
res[j] = BASE_COEFFICIENTS(Derivative, j) * ti;
ti = ti * t;
}
}
}
static const MatN
BLENDING_MATRIX; ///< Blending matrix. See \ref computeBlendingMatrix.
static const MatN BASE_COEFFICIENTS; ///< Base coefficients matrix.
///< See \ref computeBaseCoefficients.
Eigen::aligned_deque<SO3> knots_; ///< Knots
int64_t dt_ns_; ///< Knot interval in nanoseconds
int64_t start_t_ns_; ///< Start time in nanoseconds
std::array<_Scalar, 4> pow_inv_dt_; ///< Array with inverse powers of dt
}; // namespace basalt
template <int _N, typename _Scalar>
const typename So3Spline<_N, _Scalar>::MatN
So3Spline<_N, _Scalar>::BASE_COEFFICIENTS =
computeBaseCoefficients<_N, _Scalar>();
template <int _N, typename _Scalar>
const typename So3Spline<_N, _Scalar>::MatN
So3Spline<_N, _Scalar>::BLENDING_MATRIX =
computeBlendingMatrix<_N, _Scalar, true>();
} // namespace basalt
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