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The solution to your problem is as follows:

#include "ros/ros.h"
#include <beginner_tutorials/TwoInts.h>
#include "std_msgs/Int16.h"

ros::Publisher pub;

void callback(const beginner_tutorials::TwoInts::ConstPtr& msg)
{
    ROS_INFO("I heard: [%d, %d]", msg->a, msg->b);

    std_msgs::Int16 sum;

    sum.data = msg->a + msg->b;

    pub.publish(sum);
}

int main(int argc, char **argv)
{
  ros::init(argc, argv, "sum");
  ros::NodeHandle n;

  pub = n.advertise<std_msgs::Int16>("sum", 1000);
  ros::Subscriber sub = n.subscribe("two_ints", 1000, callback);

  ros::spin();

  return 0;
}

I believe you are confusing Int16 message with the int16 type. The Int16 message looks like this:

int16 data

The Int16 message contains a field called data that is an int16 type.

So, msg->a and msg->b get the int16 type values from the Twoints message. Now, you want to assign the result of sum of the type int16 values back into a Int16 message to publish it. This requires you to assign it to the data field.