From 9b7eb34add3f5a21fcf571c238c244476c8b15fd Mon Sep 17 00:00:00 2001
From: Frank Dellaert <dellaert@gmail.com>
Date: Mon, 15 Apr 2019 15:19:40 -0400
Subject: [PATCH] Show how expressions make (optimization-based) inverse
 kinematics easy.

---
 .../InverseKinematicsExampleExpressions.cpp   | 92 +++++++++++++++++++
 1 file changed, 92 insertions(+)
 create mode 100644 examples/InverseKinematicsExampleExpressions.cpp

diff --git a/examples/InverseKinematicsExampleExpressions.cpp b/examples/InverseKinematicsExampleExpressions.cpp
new file mode 100644
index 000000000..2ccb05c96
--- /dev/null
+++ b/examples/InverseKinematicsExampleExpressions.cpp
@@ -0,0 +1,92 @@
+/* ----------------------------------------------------------------------------
+
+ * GTSAM Copyright 2010, Georgia Tech Research Corporation,
+ * Atlanta, Georgia 30332-0415
+ * All Rights Reserved
+ * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
+
+ * See LICENSE for the license information
+
+ * -------------------------------------------------------------------------- */
+
+/**
+ * @file InverseKinematicsExampleExpressions.cpp
+ * @brief Implement inverse kinematics on a three-link arm using expressions.
+ * @date April 15, 2019
+ * @author Frank Dellaert
+ */
+
+#include <gtsam/geometry/Pose2.h>
+#include <gtsam/nonlinear/ExpressionFactorGraph.h>
+#include <gtsam/nonlinear/LevenbergMarquardtOptimizer.h>
+#include <gtsam/nonlinear/Marginals.h>
+#include <gtsam/nonlinear/expressions.h>
+#include <gtsam/slam/BetweenFactor.h>
+#include <gtsam/slam/PriorFactor.h>
+#include <gtsam/slam/expressions.h>
+
+#include <cmath>
+
+using namespace std;
+using namespace gtsam;
+
+// Scalar multiplication of a vector, with derivatives.
+inline Vector3 scalarMultiply(const double& s, const Vector3& v,
+                              OptionalJacobian<3, 1> Hs,
+                              OptionalJacobian<3, 3> Hv) {
+  if (Hs) *Hs = v;
+  if (Hv) *Hv = s * I_3x3;
+  return s * v;
+}
+
+// Expression version of scalar product, using above function.
+inline Vector3_ operator*(const Double_& s, const Vector3_& v) {
+  return Vector3_(&scalarMultiply, s, v);
+}
+
+// Expression version of Pose2::Expmap
+inline Pose2_ Expmap(const Vector3_& xi) { return Pose2_(&Pose2::Expmap, xi); }
+
+// Main function
+int main(int argc, char** argv) {
+  // Three-link planar manipulator specification.
+  const double L1 = 3.5, L2 = 3.5, L3 = 2.5;    // link lengths
+  const Pose2 sXt0(0, L1 + L2 + L3, M_PI / 2);  // end-effector pose at rest
+  const Vector3 xi1(0, 0, 1), xi2(L1, 0, 1),
+      xi3(L1 + L2, 0, 1);  // screw axes at rest
+
+  // Create Expressions for unknowns
+  using symbol_shorthand::Q;
+  Double_ q1(Q(1)), q2(Q(2)), q3(Q(3));
+
+  // Forward kinematics expression as product of exponentials
+  Pose2_ l1Zl1 = Expmap(q1 * Vector3_(xi1));
+  Pose2_ l2Zl2 = Expmap(q2 * Vector3_(xi2));
+  Pose2_ l3Zl3 = Expmap(q3 * Vector3_(xi3));
+  Pose2_ forward = compose(compose(l1Zl1, l2Zl2), compose(l3Zl3, Pose2_(sXt0)));
+
+  // Create a factor graph with a a single expression factor.
+  ExpressionFactorGraph graph;
+  Pose2 desiredEndEffectorPose(3, 2, 0);
+  auto model = noiseModel::Diagonal::Sigmas(Vector3(0.2, 0.2, 0.1));
+  graph.addExpressionFactor(forward, desiredEndEffectorPose, model);
+
+  // Create initial estimate
+  Values initial;
+  initial.insert(Q(1), 0.1);
+  initial.insert(Q(2), 0.2);
+  initial.insert(Q(3), 0.3);
+  initial.print("\nInitial Estimate:\n");  // print
+  GTSAM_PRINT(forward.value(initial));
+
+  // Optimize the initial values using a Gauss-Newton nonlinear optimizer
+  LevenbergMarquardtParams params;
+  params.setlambdaInitial(1e6);
+  LevenbergMarquardtOptimizer optimizer(graph, initial, params);
+  Values result = optimizer.optimize();
+  result.print("Final Result:\n");
+
+  GTSAM_PRINT(forward.value(result));
+
+  return 0;
+}